Finite-Dimensional Division Algebras over Fields

Finite-Dimensional Division Algebras over Fields

Nathan Jacobson

Finite-Dimensional Division Algebras over Fields

Nathan Jacobson (1910–1999) Yale University, New Haven, CT, USA Paul Moritz Cohn (1924–2006) University College London, UK

Library of Congress Cataloging-in-Publication Data

Jacobson, Nathan, 1910-1999 Finite-dimensional division algebras over fields / Nathan Jacobson. p. cm. Includes bibliographical references (p. 275-280). ISBN 3-540-57029-2 (Berlin : hardcover : alk. paper) 1. Division algebras. 2. Fields (Algebra) I. Title. QA247.45.J33 1996 512’.24--dc20 96-31625 CIP

Corrections of the 1st edition (1996) carried out on behalf of N. Jacobson (deceased) by Prof. P.M. Cohn (UC London, UK) ISBN 978-3-540-57029-5 e-ISBN 978-3-642-02429-0 DOI 10.1007/978-3-642-02429-0 Springer Heidelberg Dordrecht London New York Mathematics Subject Classification (1991): 13-XX, 16-XX, 17-XX © Springer-Verlag Berlin Heidelberg 1996, Corrected 2nd printing 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

PREFACE

These algebras determine, by the Sliedderburn Theorem. the semi-simple finite dimensional algebras over a field. They lead to the definition of the Brauer group and to certain geometric objects, the Brauer-Severi varieties. Sie shall be interested in these algebras which have an involution. Algebras with involution arose first in the study of the so-called .'multiplication algebras of Riemann matrices". Albert undertook their study at the behest of Lefschetz. He solved the problem of determining these algebras. The problem has an algebraic part and an arithmetic part which can be solved only by determining the finite dimensional simple algebras over an algebraic number field. We are not going to consider the arithmetic part but will be interested only in the algebraic part. In Albert's classical book (1939). both parts are treated. A quick survey of our Table of Contents will indicate the scope of the present volume. The largest part of our book is the fifth chapter which deals with involutorial rimple algebras of finite dimension over a field. Of particular interest are the Jordan algebras determined by these algebras with involution. Their structure is determined and two important concepts of these algebras with involution are the universal enveloping algebras and the reduced norm. Of great importance is the concept of isotopy. There are numerous applications of these concepts, some of which are quite old. In preparing this volume we have been assisted by our friends, notably Jean-Pierre Tignol and John Faulkner. Also, I arn greatly indebted to my secretary. Donna Belli, and to my wife, Florie. I wish to thank all of them for their help.

Table of Contents

I . Skew Polynominals and Division Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. Skew-polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Arithmetic in a PID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3. Applications to Skew-polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4. Cyclic and Generalized Cyclic Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5. Generalized Differential Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 1.6. Reduced Characteristic Polynomial, Trace and Norm . . . . . . . . . . . . . 24 1.7. Norm Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.8. Derivations of Purely Inseparable Extensions of Exponent One . . . .31 1.9. Some Tensor Product Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.10. Twisted Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.11. Differential Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 I1. Brauer Factor Sets and Noether Factor Sets . . . . . . . . . . . . . . . . . . . . . . . . . .41 2.1. Frobenius Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2. Commutative Frobenius Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.3. Brauer Factor Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4. Condition for Split Algebra . The Tensor Product Theorem . . . . . . . . 51 2.5. The Brauer Group B r ( K / F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.6. Crossed Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.7. The Exponent of a Central Simple Algebra . . . . . . . . . . . . . . . . . . . . . . . 60 2.8. Central Division Algebras of Prescribed Exponent and Degree . . . . 62 2.9. Central Division Algebras of Degree < 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.10. Won-cyclic Division Algebras of Degree Four . . . . . . . . . . . . . . . . . . . . . . 76 2.11. A Criterion for Cyclicity of a Division Algebra of Prime Degree . . . 80 2.12. Central Division Algebras of Degree Five . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.13. Inflation and Restriction for Crossed Products . . . . . . . . . . . . . . . . . . . . 86 2.14. Isomorphism of B r ( F ) and H 2 ( F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 I11. Galois Descent and Generic Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1. Galois Descent for Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.2. Forms of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.3. Forms and Non-cornmutativc Cohomology . . . . . . . . . . . . . . . . . . . . . . . 102 3.4. Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.5. Brauer-Severi Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Table of Contents

viii

3.6. 3.7. 3.8. 3.9. 3.10. 3.11. 3.12. 3.13.

Properties of Brauer-Severi Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Brauer-Severi Varieties and Brauer Fields . . . . . . . . . . . . . . . . . . . . . . . 118 Generic Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Properties of Brauer Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Central Simple Algebras Split by a Brauer Field . . . . . . . . . . . . . . . . . 130 Norm Hypersurface of a Central Simple Algebra . . . . . . . . . . . . . . . . . 138 Variety of Rank One Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 The Brauer Functor . Corestriction of Algebras . . . . . . . . . . . . . . . . . . . 149

IV . p-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.1. The Frobenius Map and Purely Inseparable Splitting Fields . . . . . . 155 4.2. Similarity to Tensor Products of Cyclic Algebras . . . . . . . . . . . . . . . . 158 4.3. Galois Extensions of Prime Power Degree . . . . . . . . . . . . . . . . . . . . . . . . 162 4.4. Conditions for Cyclicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 4.5. Similarity to Cyclic Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.6. Generic Abelian Crossed Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 4.7. Non-cyclic p-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

V . Simple Algebras with Involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.1. Generalities . Simple Algebras with Involution . . . . . . . . . . . . . . . . . . . . 186 5.2. Existence of Involutions in Simple Algebras . . . . . . . . . . . . . . . . . . . . . . 193 5.3. Reduced Norms of Special Jordan Algebras . . . . . . . . . . . . . . . . . . . . . . 197 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10. 5.11. 5.12. 5.13.

Differential Calculus of Rational Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Basic Properties of Reduced Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Low Dimensional Involutorial Division Algebras . Positive Results 209 Some Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Decomposition of Simple Algebras with Involution of Degree 4 . . . 232 Multiplicative Properties of Reduced Norms . . . . . . . . . . . . . . . . . . . . . 235 Isotopy and Norm Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 Special Universal Envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Applications to Norm Similarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 The Jordan Algebra H ( A . J) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

I. Skew Polynomials and Division Algebras

MTeassunle the reader is familiar with the standard ways of constrlicting "simple" field extensions of a given field F . using polynomials. These are of t,wo kinds: the simple transcendental extension F ( t ) , which is the field of fractions of the polynomial ring F[t] ill an indeterminate t; and t,he simple algebraic extensioil F[t]/(f (t)) where f ( t ) is an irreducible polyilomial in F[t]. In this chapter we shall consider some analogous constructions of division rings based on certain rings of polynomials D[t; a: S] that were first introduced by Oystein Ore [33] and simultaneously by Wedderburn. Here D is a given division ring. a is an automorphism of D , 6 is a a-derivation (1.1.1) and t is an indeterminate satisfying the basic cornrnutation rule

for a E D. The elements of D [ t ;0 , 61 are (left) polynomials

where multiplication can be deduced from the associative and distriblitive laws and (1.0.1) (cf. Draxl [83]).Lie shall consider two types of rings obtained from D [t; a. 61 : homomorphic images and certain localizatiorls (rings of quotients) by central elements. The special case in which 6 = 0 leads to cyclic and generalized cyclic algebras. The special case in which o = 1 and the characteristic is p # 0 gives differential extensions arlalogous to cyclic algebras. The rings D[t; a. 61 are principal ideal domains, that is. they are rings without zero divisors in which all one-sided ideals are principal. We shall develop the necessary arithmetic of such domains and use this to derive results on cyclic and generalized cyclic algebras and their differential analogues.

1.1. Skew-polynomial Rings Let R be a ring (with 1 and the usual coilventions on homomorphisms and subrings of unital rings), a a ring endomorphism of R. S a left a-derivation of R , that is, S is additive and for a, b, E R )

6(ab)= (aa)( 6 b )

+ (6a)b

N. Jacobson, Finite-Dimensional Division Algebras over Fields , © Springer-Verlag Berlin Heidelberg 1996, Corrected 2nd printing 2010

(1.1.1)

2

I. Skew Polynomials and Division Algebras

which implies S(1) = 0. Let R[t;a! 61 be the set of polynomials

where the a, E R and equality and addition are defined as usual. In particular. t is transcendental over R in the sense that a0 alf . . . antn = 0 + a, = 0. 0 5 2 n . Evidently. R[t: a , & ]is a free (left) R-module (with the obvious nlodule structure). We wish to make R[t; a,S] into a ring in which we have the relation ta = ( a a ) t ha, a E R. (1.1.3)

+

<

+ +

+

Then associativity and the distributive laws imply that

where ST13satisfies the recursion formula

and Soo = l R (identity map); Slo = 6: Sll = a! by (1.1.3). It follows that S,,,; 0 5 j 5 12: is a sum of all the monomials in a and S that are of degree j in a and of degree n - j in 6:e.g.,

We now define

n

a(Snjb)tJITn

(at")(btm) = 3 =o

where S,, is defined by (1.1.5) and Soo= lR.and we define products of poll-nomials in t bv this and the distributive laws:

To see that R[t:a. S] is a ring it suffices to check the associative law of multiplication. A dlrect verification of this is rather tedious. We shall prove associativity by using a representation by infinite row-finite matrices with entries in R We denote the set of matrices whose rows are infinite sequences of elements of R with only a finite number of nonzero entries in each row by MW(R).It is ~vellknown and readily verified that this is a ring under the usual matrix compositions For u E R we define

1.1. Skew-polynomial Rings

3

and

wherc e,, is t h e matrix i n n / f d ( R )w i t h 1 i n (z,j)-positionand 0's elsewhere. Using (1.1.5) we can prove b y induction o n n t h e Leibniz formula

where we take S n j = 0 i f j > n. T h i s formula implies that a --, a' is a nlonomorphism o f R ont'o a subring R' o f M U ( R ) .Direct verification shows also that t'a' = ( a a ) ' t l (Sa)'. T h i s implies that (alt'")(b't'") = CyzOa'(Snjb)'t'J-m. I t follows that C a j t j --, Cait'j is a homomorphism o f R[t:a ) S] into h f u ( R ) . It is readily seen also that Ca;t1J= 0 o a; = 0 , j 2 0 . T h i s implies that we have a rnonomorpllism o f R [ t ;a , S] into Il.fw ( R ) and hence that R [ t ;rr, S] is a ring. called a ske~rrpoly~aorninl ring. From now o n we assurrle R = D is a division ring. T h e n CJ is a monomorpllisnl. I f f ( t ) = no a l t . . . antn wit,h a , # 0 we define deg f ( t ) = n. .Also, we put deg 0 = -m. I f g ( t ) = bo blt . . . bmtm, b,, # 0 , t h e n ,f ( t ) g ( t )= . . . + a , ( a nbm)t7L+rnand a , ( C J , b,,) # 0 . Hence

+

+

+

+

+

deg f g = deg f

+

+

+ deg g.

(1.1.10)

T h i s implies t h a t D [ t ;a . S] is a domain. that is, has n o zero divisors # 0 W e r5tablish next a left division algorithm i n D [ t :n, S ] , that fol any f ( t ) ,g ( t ) E D [ t ;CJ.S] w i t h g # 0 there exist unique q ( t ) .r ( t ) w i t h deg r ( t ) < deg g ( f ) such that (1.1.11) f ( t ) = q ( t ) g ( t )+ r ( f ) . Suppose J ( t ) = a0 + a l t + . . . + a n t n , g ( t ) = bo + blt + . . . + b,tm b , # 0 . I f n < nl we have f ( t ) = Og(t) f ( t ) . and i f n rn, we have

+

f ( t )- a71(an-nzbm)-1t'z-mg(t) = ai_,tnP1

>

+ ....

where (1.1.12)

Hence t h e existence o f q ( t ) and r ( t ) follow b y induction o n n. T h e uniqlieness fhllow b y degree considcrations. A ring R is called a left (right) principal ideal domain (abbreviated as left, or right PID) i f every left (right) ideal in R is principal, that is, has t h e form R,a (aR). T h e existence o f t h e left division algorithm in R = D [ t ;a . 61 implies i n t h e usual way that R is a left PID. A ring R is called left noethe'rian i f it satisfies t h e ascending chain condition for left ideals. R is said t o satisfy t h e left Ore- Wedderburn condition i f given a E R and s regular i n R (that is not a zero divisor) there exist a1 E R, sl regular i n R such t h a t s l a = a l s . For a domain t,his is equivalent t o : R a n R b # 0 for any a # 0 ; b # 0 .

4

I. Skew Polynomials and Division Algebras

Proposition 1.1.13. If R is a domain we have the followin,g implications: (i) R is a le,ft PID jfii) R is left noetherinn j(zii) R satisfies the left Ore- Wedrlerburn, condition. Proof. (i) + (ii) is clear since (ii) is equivalent to the condition that every left ideal is finitely generated. To sllow that (ii) 3 (iii), let a and b be non-zero elenleiits of R. We have the ascending chain of left ideals Ra

c R a + Rab c R a + Rab + ~ a b c ' ....

+

+ +

Hence we have Ra Rab . . . ~ k . Then we have :c, E R such that

Not all the 2, = 0. Let xi, be the first with zh f 0. Cancelling bh we obtain

+

a =bRa ~ Rub

+ . . . + Rabk+'

for some

# 0. Then abk+l = xhubh + . . . + xkabk

and the Ore-Wedderburn condition holds. This condition insures that R can be embedded in a (left) quotient division ring Q(R) (or QL(R)).that is. a division ring containing R as subring such that every element of Q has the form b-'a, b,o E R . We shall not give the proof since the theorem will play only a marginal role in the sequel (see Section 1 11). An interested reader may consult Jacobson [43]. p. 118f or ex. 6 on p. 119 of BAI for a proof. Conditions (ii) and (iii) hold for R = D [ t ;n, S] since this is a left PID. On thc othcr hand. we have

Proposition 1.1.14. R = D[t;a,S] Is a righ,t PID if and only if a is an automorphism. Proof. If a is an automorphism we have the relation

which implies that if we put a / = n-' and 6' = -&ap1 then

+

hl(ab) = a(S1b) (S1a)(a'b)

(1.1.16)

(cf. 1.1.1) Moreover. every element of R can be written m one and only one way in the form no tal t2a2 . . . tna,, a, E D . It follows by symmetry that we have a right division algorithm and hence that R is a right PID. Conversely. suppose R is a right PID and let a E D* = D\{O} (\ denotes 5et theoretic complement). By the right-handed version of Proposition 1.1.13. R has the right Ore property. Hence t R atR # 0 and so we have f (t),g(t) # 0 such that t f (t) = atg(t). Then deg f (t) = deg g(t) and f (t) = ao a l t ..

+

+

+ +

n

+ +. +

5

1.l. Skew-polynonlial Rings

antn. g ( t ) = bo + blt + . . . + bntn with an highest degree we obtain a a , = arb,,

# 0, b, # 0. Comparing terms

of

a = aa,bil

Thus a E O D . Since a was any non-zero element of D , we see that a is surjective and hence a is an automorphism.

\lVe sliall call D [ t :a , 61 a differential polynomial ring if a = 1 and a twisted polynomial ring if 6 = 0 or, more generally, if these situations can be realized by replacing t by another generator for R over D of the form t' = u t + v , u E D * , .c E D . Then t = u ' t ' t v ' where u' = u-I: v' = - 1 1 - l ~ and t'a = (a1a)t'+6' where (1.1.17) a'a = u ( a a ) u p l , 6'a = u ( 6 a ) va - u ( a a ) u - l v .

+

The first of t,hese equations shows that if a is an inner automorphism then we may take c' = 1 so we have a differential polynomial ring. The second equation ill (1.1.17) shows that if we have 6 = 0 t,hen replacing t by t' = t v gives the new rr-derivation

+

which we call an inner a-derivation. Also we see that if 6 has this form then replacing t by t' = t - v gives 6' = 0 so we have a twisted polynomial ring. We therefore have

Proposition 1.1.19. If a is arL inner automorphism, then R = D [ t ;a , 61 is a diflerential polynomial ring and if 6 is an inner a-derivation then R ,is a twisted polynomial ring. From now on we assume that a is an automorphism. This implies that the center C of D is stabilized by a and we have the following

Proposition 1.1.20. If the restriction alC mial ring.

#

lc then R is a twisted polyno-

Proof. By hypothesis, there exists a c E C such that ac # c so we can replace = ct - t c = ( c - a c ) t - 6c. This replaces 6 by 6' as in (1.1.17) where u = c - ac and ,u = -6c. Then for any a E D ,

t by t'

+

6'a = ( c - 6c)6a - (6c)a (aa)Sc (since u E C ) = c(6a) + ( a a ) ( 6 c )- ( a c )(ha) - (&)a = S(ac) - 6(ca) = 0 Hence R is a twisted polynomial ring.

Theorem 1.1.21. If the dimensionality [ D : C ] < oo then R = D [ t ;a , 6 ] is either a twisted polynomial ring or a differential polynomial ring.

6

I. Skew Polynomials and Division Algebras

Proof. If a I C # l C then R is a twisted polyilonlial ring by 1.1.20. On the other hand, if cr I C = l c then a is an inner automorphism by the SkolemNoetller theorem, Then R is a differential polyrlomial ring by 1.1.19. Since we are assuming o is a n automorphism, R = D [ t ;a. 61 is both a left and a right PID. A ring of this sort will be called simply a PID. \;lie now consider the (two sided) ideals in any PID R. If I is such a n ideal, then I = R d = d * R and for any a E R there exist a', (1 E R such that da = a'$; ad* = d*&. Also d = d * u , d* = vd for u , 2; E R and d = v d u = vu'd. Hence UU' = 1 and since R is a domain u'v = 1 (by ( u ' v - 1)u' = 0). Hence v is a unit. Similarly u is a unit. Then d R = d * u R = d * R = I so we have I = R d = dR.Conversely, if d is an elernent such that for any a E R there is a n a' and a n Li such that da = a'd and ad = dFi then R d is an ideal and Rd = d R . An elernent having this property will he called a two-sided element of R. It is readily seen that a -i a' is a n automorphism of R with a -i 6 as its inverse. It is easily seen also that if d is a two-sided elernent, then so is u d v for any units u and 11 and if d l = dads and any two of these are two-sided, t,hen so is the third. T;CJe shall now determine the t'wo-sided elements, hence, the ideals in twisted polynomial and differential polynomial rings. We assume first that R = D [ t ;a ] = D [ t ;a, 01.

Theorem 1.1.22. ( i ) T h e two-sided elements of R = D [ t ;a] a w the elements a c ( t ) t r Lwhere a E D , n = 0, 1. ... and c ( t ) E Cent R, the center of R. (ii) Cent R = Cent D n I n v ( a ) , where Inv(cr) = { d E D I ad = d } , if n o non-zero power of a i s a n i n n e r automorphism of D . (iii) Let a have finite order r m,odulo innwr automorph~ismsand su,ppose a7'= Iu : x --i U L U - l , u E D ' . T h e n Cent R Is the set of polynom,ials of the form 7 0 + r l u p l t r + rzu-2t2r+ , . . + y s u p s t S T (1.1.23) where -/, E Cent D and Y,u-' E Inv(a). hforeover, zf r zs also the order of I Cent D t h e n u can he chosen zn Inv(a) a r ~ dthen Cent R = F[7~-'t'] tlze r ~ n gof polynomzals zn u - ' t r wzth coeficzents zn F = Cent D Inv(cr). T h e last sztuatzon holds zf [ D : Cent D ] < cc.

a

n

Proof. (i) The elements t n are two-sided and any elerlient of D is two-sided. Hcnce any two-sided element has the form ac(t)t7' where a E D and c ( t ) is a two-sided element of the form c(t) = 1

+ clf + c2t2 + . . + cmtriL,ci E D : c,, ,

# 0.

(1.1.24)

The conditions that c ( t ) is two-sided are that for every a E D there exists a n a' E R sucll that c ( t ) a = n l c ( t ) and there exists a t' E R such that c ( t ) t = t f c ( t ) .It follows that a' E D and then that a' = a . Also t' = t . Hence c ( t ) E Cent R. (ii) Me have Cent R = Cent D n Inv(cr). Let F denote this field. If Cent R 2 F then Ccnt R contains an elernent c,tz of degree m > 0.

7

1.1. Skew-polynomial Rings

Then every c,ti E Cent R. In particular, c , t m f 0 is in Cent R. Then am,a = c,'acm, a E D , so a m is inner. (iii) Let be the order of a mod1110 inner automorphisms and let a r = Iu. Then m = 7s and amlc: = c,lxcm = uS2u-S implies that c, = ? , y u p S , y sE Cent D. A similar argument applies to every citZ # 0 in Cent R. This implies that any element of Cent R has the form (1.1.23) with ;!, E Cent D . The condit,ion that such an element commutes with t is %?L-JE Inv(a). We have a r z = uxu-' and o a r = o r a so ( a u ) ( a x ) ( a u ) - ' = u ( a x ) t ~ - ' . Hence a u = pu where p E Cent D. Then a 2 u = ( a ~ ~ ) and p u u = uuu-' = a T u = ( a r - ' p ) . . . ( a p ) p u SO ( o r - ' p ) . . . ( o p ) p = 1. If r is the order of a / Ccnt D then. this reads Ncent D , F ( p ) = 1 where A' is the norm. Hence, by Hilbert's norin theorem ("Satz 90") there exists a X E Cent D such that /* = X ( O X ) ~ 'We . may replace u by Xu and so we may suppose that a u = u . Tllerl also ayi = yi and Cent R = F[u-'t']. Tlle la,st statenlent follows from the Skolcm-Noether Theorem.

-

S;lTesuppose next that R = D [ t ;S ] D [ t ;1 , 6 ] .We recall that the set of 6-constants, t>hat is, t,he a E D such that Sa = 0 form a division subring, Const 6:of D . We recall also that if y E Cent D then yS is a derivation and if t,he characteristic char D = p # 0 then 6P is a derivation. It follows in the characteristic p case that if the yi E Cent D then

is a derivation. We note also that as a special case of (1.1.4) we have

Hence, if c ( t ) = Cg cjt3 then

\Ve now define a D-linear transformation A,, z = 0 . 1 , . . . in D [ t :61 by

<

Then A,c(t) = C g ,

~ , ? t ' -if~z n and A,c(t) = 0 if i > n,. If char D = 0 (j) then A , c ( t ) = $c(')( t ) where c(" ( t ) is the formal i-th derivative of c ( t ) . For arbitrary characteristic, we can verify that A k A 2 t j = ( k

dkai= ( k

: a,,, i,

=

n,n,.

:

i, Akiitj. Hence

(1.1.29)

8

I. Skew Polynomials and Division Algebras

+

) a~ t D and Since 6 is a derivation in D we have aL6 = 6aL ( 6 ~ for a~ the left nlultiplication x --t a x . Since a --t a~ is a ring homomorphism tile relation we have noted implies that we have a uiiique hoinomorphism of R into End D such that a --i a ~ t , --i 6. TVe denote the image of f ( t ) under this hoinomorphism bv f L ( 6 ) and we abbreviate f L ( 6 ) a to f ( 6 ) a . Using this notation and the definition of the A, we call write (1.1.27) as

WP can now prove Lemma 1.1.31.

(i) If c ( t ) = C: c,tZ, ci E D , and [ c ( t ) a] , = 0 ,for n E D then [ A k c ( t ) a] , =0 for all k . jii) If c ( t ) E Cent R then every A k c ( t ) E Cent R. Proof. (i) B y (1.1.30), [ c ( t ) a] , = 0 if and only if a c ( t ) = C i ( A , c ) ( ~ ) a tApply t Ak for 0 < k 5 n to both sides of this relation. This gives

which shows that A k c ( t ) satisfy the condition [ A k c ( t ) a] , = 0. Since A k c ( t ) = 0 for I; > n we obtain (i). (ii) c ( t ) E Cent R if and only if [ c ( t ) a] , = 0 for all a E D and [ c ( t ) t] , = 0. The last condition holds if and only if Sc, = 0 for all z. It is now clear that (ii) follows from (i). We can now determine the two-sided elements of a ring of differential polynomials.

Theorem 1.1.32 (Amitsur [ 5 7 ] ) . fi) T h e two-sided elements of R = D [ t ;61 are the elements u c ( t ) where u E D and c ( t ) E Cent R. (ii) Either Cent R = F = Cent D Const 6 o r Cent R = F [ z ] where z has the following form

n

t-d z={tpe+71tpe-'+..-+7,t-d

if char D = 0 if c h a r D = p

(1.1.33)

where i n th,e J r s t case 6 = id the inner derivation n: --i [ d ,x] and i n the second case the y, E F, Sd = 0 and

1.1. Skew-polynomial Rings

9

Proof. (i) The condition that an element is two-sided shows that any inonic two-sided element is in the center. This implies (i). (ii) IIJe have Cent R n D = F and this is a proper subset of Cent R if and only if Cent R contains elements of positive degree. Let c ( t ) = co cl . + c,tn be 5uch an elelneilt of least positive degree n. BY ( 1 1.30). every

+ +

A,c(t)

E

15j

0,

Cent R. By the minimality of the degree of c ( t ) lie have


(;)

c,

=

~,

-

1: i > 1. Since the binomial coefficients ( . l < j < i - l ,

are 0 in D if and only if char D = p and i = pe we see that if char D = 0 then c ( t ) = co+clt and if char D = p then c ( t ) = co+clt+c,tp+cpztp2+...+cpetp e . In bot,h cases colnmutativity with t implies that the ci E Const 6. If c ( t ) = co c l t then 0 = [ c ( t ) o] . = [co,a] [el,a ] t el (6a). Hence cl E Cent D so el E F. Then we may assurrle cl = 1 and c ( t ) = t - d . Then 6a = [d;a ] . If cp3tp3 t,hen [tp3 a] = 6" a (by (1.1.26)) and char D = p and c ( t ) = co hence e e

+

+

+

+

+

0 = [cg;a] x [ c p , , a ] t p + 3 ):C~~(W~~).

X:

Then c,, E F and cp36p3is the inner derivation z -i [d.z ] , d = - c c IVe may normalize c ( t ) to c ( t ) = tpe rltpe-' . . . + ~ , ~ dt . E F . and we have (1.1.35). We now write z = t-d if char D = 0 and z = t"'flltpe-'+. . iet-d ~f char D = p. It remains to show that Cent R = F [ z ] . Since F C Cent R and z E Ccnt R, F [ z ] c Cent R. Now let f ( t ) E Ccnt R. By division we obtain

+

f

+

.+-

(1.1.35)

( t ) = q ( t ) zf r ( t )

u~heredeg r ( t ) < deg z . We claim that q ( t ) ,r ( t ) E Cent R. For we have 0 = [f(t),t= ] [ q ( t ) . t ] z + [ r ( t ) , and t] 0 = [ f ( t ) , a= ] [ q ( t ) . a ] z + [ r ( t ) , aa] ,E D . Degree considerations show that [ q ( t ) t] . = [ q ( t )a] , = [ r ( t ) t. ] = [ r ( t ) a] , =0 and hence q ( t ) .r ( t ) E Cent R. We can now use induction on the degree of f ( t ) to conclude that f ( t ) E F [ z ] Hence F [ z ]= Cent R. The foregoing result implies that if char D = 0 then Ccnt R = F unless 6 is an inner derivation, and if char D = p then Cent R = F unless there exist,s a monic p-polynomial f ( A ) = Ape ylApe-l . . . ?/eX with :i, E F such that f ( 6 ) = id where 6d = 0. Moreover, these conditjioi~sare sufficient for Cent R 2 F. For in the first case if 6 = id then t-d E Cent R and in the second case if f ( 6 ) = id with 6d = 0 then t p e ?ltpe-l . . . ?,t - d E Cent R. Moreover, if z is chosen as in the proof then the corresponding polynomial ,f ( A ) = Ape y l ~ l - ' e - l . . . is the monic polynomial of least degree such that f (6)is an inner derivation by a d such that 6d = 0.

+

+

+

+ +

+ +

+ +

10

I. Skew Polynomials and Division Algebras

1.2. Arithmetic in a PID Let R be a PID (= left and right PID). SVe shall work with left ideals R a and the corresponding factor R-modules R j R a . By symmetry. the results will apply equally well to right ideals. Suppose R a > R b # 0. Then b = ca so a is a right factor of b. We indicate this by writing a 1. b. Conversely. if a 1,. b then R a > Rb. This corlditiorl implies that R a j R b is a submodule of R j R b . Now R a l R b is cyclic with generator a + Rb. It is clear that the annihilator of this generator is R c . Hence (1.2.1) R a l R b = R a l R c a -. R / R c . JVe also have

( R / R b ) / ( R a / R b ) -. R / R a .

(1.2.2)

SVe h a ~ ~ R ae = R b f 0 if and only if a . , 1 b and b ,1 a . Then b = c a , a = db so b = cdb. Then cd = 1 which implies also that dc = 1 since R is a domain. Thus c and d are units. Hence a and b are left associates in the sense that 11 = u a : u a unit. J4'e have R a R b = R d . Then d 1, (1 and d ,1 b. Moreover, if e . ,1 a and e 1,- b then R e > R a and R e > R b so Re 3 R d and e 1,. d. Hence d is a right greatest c o m m o n divisor (right g.c.d.) of a and b in the obvious sense. Any two right g.c.d. are left associates. JVe denote any right g.c.d. of a and b (= any- d such that R,a R b = R d ) by ( a , b),.. \Ve have seen that R satisfies the left Ore condition. If a # 0 and b # 0 then R a n R b # 0.We have R a n R b = Rm so m = b'a = a'b # 0. Moreover, if a 1,. n a n d b 1,. n then Rm = R a n R b > Rn so m -,1 n . Hence m is a left least c o m m o n multiple (left 1.c.m) of a and b in the obvious sense. Any two left 1.c.m. of a and b are left associates. SVe denote any one of these by [a.b]!. We have seen that R is left noetherian. JVe now show that R is left artinian rriodulo any non-zero left ideal R a , which means that if we have a sequence of left ideals

+

+

Ral

> Raa > . . . > R a # 0

then there exists a k such that R a k (1.2.3) is equivalent to

=

(1.2.3)

R a k + l = . . . . To see this we note that

Then a = b,a, = b,+la,+l = b,+lc,a, so b, = b,+lci and

Since R is right noetherian we have b k R = b k + l R = . . . for some k . Then c k , cn+l,. . . are units and R a k = Rak+l = . ... The condition that R is left artinian modulo any non-zero left ideal R a is equivalent to R / R a is artinian for any a # 0. Now we recall that a module has a composition series if and only if it is both artinian and noetherian. Hence we have

1.2. Arithmetic in a PID

11

Proposition 1.2.6. The left R-module R/Ra has a composition series if a # 0. T;lTedetermine next an element condition that R/Ra b # 0.

-. R/Rb for a #

0.

Definition 1.2.7. If a and b are non-zero elements of R then a is said to be b ) if there exists a u in R such that left szmilar to b ( a

( u ,b ) , = 1 and a = [b,u ] ~ u - ' . The condition that

(TL,

b ) , = 1 is equivalent to the existence of x and y such 1 =xu+yb

and a = [b.T I ] ~ uis - ' equivalent to

m = u'b = au = 1. Thus we have a u' such that (u'.a ) [ = 1 and b = where (u'. ti',a],. Hence if a is left similar to h then b is right similar to a. Similaritv is an equivalence relation since we have

Proposition 1.2.8. Ifa and b are n,on,-zeroelements of R then R/Ra i f and only if a - p 0.

-. R/Rb

Proof. Suppose 0 is an isomorphism of R/Ra onto R/Rb and let 0 ( 1 + Ra) = IL Rb. Then

+

+

8(x + Ra) = Q ( z ( 1 Ra)) = xQ(1+ Ra) = xu

+ Rb.

+

+

Since Ra is the annihilator of 1 Ra. Ra is the annihilator of u Rb in R/Rb. Thus au E Rb and if a'u E Rb then a' = xu. Hence m = au = [u.bIe, and a = [ I L , bIeu-'. Next we note that since 0 is surjective we have an x such that xu + Rb = 1 + Rb. Then we have a y sllch that r u yb = 1 and hence ( u ,b ) , = 1. Thus a -g b. Conversely. suppose a --p b so a = [b,u]cu-land (u,b ) , = 1. Then nu = [b. u ] ~\~liich ~ implies that x E Ra H xu E Rb It follo~vsthat 8 : x Ra --i xu+Rb is well defined and is an R-n~onornorphisinof R/Ra into RlRb. This is = l+Rb. surjective since we have x, y such that I = xti+yb and hence @(x+Ra) Hence R/Rn -. R/RO.

+

+

The result a N P b + 11 N 7 . n noted above, the symmetry of a -t b and b -,a and 1.2.8 imply that R/Ra -. R/Rb ==+ R/aR -. R/bR. W e can iiow write a b for a - p b and call this equivalence relation similarity. An element p of R is called irreducible (or an atorn) if it is not a lirlit and it has no proper factors, that is, its only factors are associates and units. This is equivalent to: Rp is a maximal left ideal # 0 in R. Vlre call now prove

-

12

I. Skew Polynomials and Division Algebras

Theorem 1.2.9. The fundamental theorem of arithmetic in a PID. If R Rs a P I D any element a f 0 and not a unit of R can be written as a = p1p2 . . . p s where the p, are irreducible. Moreover. if a = p l p z . . . p , = pip; . . . p i where the p, and pi are irrehcible then s = t and there exists a permutation ( I 1 , . . , s f ) of ( 1 , .. . s ) such that p, -- p/,,.

.

Proof. W e have a composition series for R I R a . This has t h e form

and every factor module ( R a t - 1 / R a ) / ( R a t / R a ) is irreducible (or simple). T h e n R > R a l > . . . > R a , = R a and every R a , - l / R a , is an irreducible module. IVc have al = pl , a, = p,a,-1 and R I R p , R a t - l / R a , is a n irreducible module and hence p, is a n irreducible element. W e have a = a, = p,p,-1 . . . pl. Conversely. i f a = p,p,-l.. . p l where t h e p, are irreducible t h e n we let a , = p,. . . p l and we can re-trare t h e steps t o show that (1.2.10) is a composition series for R I R a . T h e second statement o n t h e lirliqueness ( u p t o permutation and similarity) o f factorizations into irreducible elements now follows frorn t h e Jordan-Holder theorem.

--

W e define t h e length E(a) o f a t o b e t h e number s o f irreducible factors p, i n a factorization a = pl . . . p , into irreducibles. Since this is t h e length o f a conzposition series for R I R a . it is clear that similar elements have t h e same length.

Proposition 1.2.11. If a and b are non-zero and non-un%tsthen l(ab) = i ( a ) Proof. Since R

+ i ( b ) = & ( [ ab]e) , + t ( ( a ,b),).

(1.2.12)

> R a > ( R a n R b ) = R [ a ,b]e we have

+

! ( [ a ,b]!) = & ( a ) & ( R a / ( R an R b ) )

(,1.2.13)

wherc t h e second & denotes t h e length o f a composition series for t h e indicated module. Now R u / ( R a n R b ) = ( R a R b ) / R b and since R > ( R a R b ) = R(a. b ) , > R b we have

+

+

+

l ( R a / ( R an Rb)) = l ( ( R a Rb)/Rb) = & ( R / R b )- ! ( R / ( R a

+ Rb))

= l ( b ) - ! ( ( a , b),).

Substitution o f this i n (1.2.13) gives (1.2.12). \Vc now consider t h e factorization theory o f (two-sided) ideals or equivalently o f two-sided elements a" ( R a * = a * R ) . Suppose a* and b* are t w o such clements and Ra' > Rb* or b* = ca* (= a * c f ) .I t follows t h a t c = c" is twosided and Rb" = ( R c * ) ( R a * ) .W e shall call a two-sided element p- two-s7ded

1.2. .4rithmetic in a PID

13

maximal (t.s.m.) if Rp" is a non-zero maximal ideal of R, or: equivalently. p* # 0 and R I R p * is a simple ring. Lemma 1.2.14. A n y ideal

# 0. # R is a product of maximal ideals.

Proof. If a* is two-sided and Ru" # 0. # R then the noetherian property of R (or Zorn's lemma) implies there exists a maximal ideal R p ; > R a * . Then a* = b*pT where b* is two-sided and Ra* = ( R b * ) ( R p ; ) so Rb* 2 R a * . If Rb' f R we repeat the process to obtain a maximal ideal R p z > Rb*. Then Rb* = ( R c W ) ( R p ; )# Rc' so Rc* 2 Rb*. Then Rn* = ( R c * ) ( R p z ) ( R p T )and R a t 2 Rb*. Continuing this waj- we obtain the result using the noetherian property.

Lemma 1.2.15. A n y m,azimal ideal is prime; that is, i,f Rp* is maximal and R p v > ( R a * ) ( R b * )where a* and b* are two-sided then either Rp" > Ra* o r Rp* > R b * .

+

Proof. Suppose Rp* 2 R a * . Then Rp* Ra* = R and Rb* = ( R p * ) ( R b e )+ ( R n w ) ( R b *C ) R p * . Similarly, Rp' 2 Rb* + Rp* > R a * . Lemma 1.2.16. A n y two maximal ideals R p * : Rq* commute. Proof. The result is clear if Rp* = R q * . Hence suppose Rp" f R q * . Then we claim that ( R p * ) ( R q * ) = Rp" n R q * . Since Rq* > Rp* n R q * , Rp* n Rq* = ( R a * ) ( R q * ) and since Rp" > Rp" n Rq* = ( R a * ) ( R q * )Rp* , > Ra" by 1.2.15. Then ( R p X ) ( R q * )> ( R a S ) ( R q * )= Rp" il R q " . Since the reverse inequality is clear, we have R p * n R q x = ( R p x ) ( R q * ) B . y symmetry R p * n R q * = ( R q " )( R p * ) . Hence ( R p " )( R q * ) = ( R q * )( R p * ) .

A consequence of Lemrnas 1.2.14 and 1.2.16 is that the set of ideals # 0 of R is a commutative monoid under multiplication, with R as unit. This monoid satisfies the cancellation law. the divisor chain and primeness condition of BA I , p. 144. As a consequence we have T h e o r e m 1.2.17. The nonzero ideals of a PID constitute a commutative monoid that i s factorial.

A n alternative form of the result is T h e o r e m 1.2.17'. If a* is a two-sided element of a PID R and a* # 0 and a* is n o t a u n i t t h e n a* = pfp; . . .p& where pd is a t.s.m. element. U p t o order and u n i t multipliers such a factorization is unique. We consider next factorizations of t.s.in. elements p" into irreducible elements of R and the structure of the corresponding simple rings R l R p * . We recall the definition of t'he idealizer of a left ideal I of a ring R. This is the

14

I. Skew Polynomials and Division Algebras

set

B = { b e R Ib

c I).

Tlle idealizer is a subring of R and it is the largest subring in which I is contained as an ideal. h'Ioreover, we have a canonical anti-isomorphism of BII into EndRR/I: the endomorphism ring of the module RII. This maps b I,b E B , int,o the endomorphism z I --i xb I ( B A 11: p. 199). UTenote next that if a* = . . pn is a factorization of a two-sided element into irreducibles then the R-modules R/Rpi are isomorphic to quotients of submodules of RIRa*. Hence these are annihilated by a* and so can be regarded as irreducible RIRa* modules. We can now prove

+

+

Theorem 1.2.19. Let p'

+

be a t.s.m. element of a P I D R. T h e n

-

( i ) T h e irreducible factors p, of arzg factorization p* = p l p z . . .p, into irreduczbles are all similar. (ii) RIRp* is a simple artinian ring which is isomorphic to a matrix ring A/r,(A) where A B,/Rp, and B, is the idealizer o,f Rp,. Proof. We have seen that R/Ra is artinian if a # 0. Hence RIRp* is simple artinian. It is well known that any two irreducible modules for such a ring are isomorphic. In particular this holds for the modules R/Rp,, which implies the similarity of any two of the p,. Since p* = plpz . . . p , : R,/Rp* has a composition series of length n . Hence R/R,p* is a direct sum of r~ left ideals isomorphic to R/Rp,. It follows that R/Rp* is anti-isomorphic to !VIn(EndRR/Rpi). Since Endfi R/Rp, is anti-isomorphic to B,/ Rp, we have R/RpX iVfn,(A),A B,/Rp,.

-

--

1.3. Applications t o Skew-polynomial Rings We consider the polynomial ring R = D [ t ;g.S] where n is an automorphism and llence R is a PID. If f ( t ) E R we have the left R-module RIRf which by restriction can be regarded as a vector space over D . If f ( t ) # 0, by the division process. R / R f has a base over D consisting of the cosets of the elements f Z , 0 i < deg f . Hence R / R f has dimension n = deg f as vector space over D . E ~ i d e n t ~ifl yR I R f and RlRg are isomorphic as R-modules they are isomorphic as D-modules. It follows t'hat if f and g are similar then they have the same degree. The module argument used to prove (1.2.12) can be applied to the various modules regarded as vector spaces over D. This yields the following result on degrees.

<

Proposition 1.3.1. If f and g are non-zero elements of R = D [ t ;CT!

S] t h e n

1.3. Applications t o Skew-polynomial Rings

15

RTenote also that the argument establishing the connection between isomorphism of the modules R/Rf and R/Rg and similarity shows that the element u in the definition of similarity can be chosen so that deg u < deg f = deg g. TVe apply this to obtain a conditiorl for similarity of t - a and t - b, o. b E D . This is the case if and only if there exists a u # 0 in D such that

for u' t R. Then u' E D and (uu)t

+ 6u

-

au = u't

-

u'b. Hence u' = a u and

The relation between n and h defined by (1.3.3) is an equivalence. If it holds for some u E D " we say that a and b are ( a , b)-conju,gute and in the special case in which 6 = 0 we use the term a-corzjugnte. and if a = 1 we use 6-conjugate. Finally. if a = 1 and 6 = 0 so we have the usual polynorrlial ring D[t] then the relation is the usual corijugacy. Our result is that t - a and t - b are similar if a.nd only if a and b are (a,6)-conjugate. We suppose next that a = b and consider the ring B / R ( t - b) where B is t,he idealizer of R(t - b). If f E B, f R ( t - b) = 71. R ( t - b) where u E D n B. Hence B / R ( t - b) is isomorphic to t,he division snhring D b of elements u E D for which there exists a u' t D such that (t - b)71 = ul(t - b). Then u' = c ~ u and the condition on IL is 61~ = bu - (au)b. (1.3.4)

+

+

We now consider the case of a twisted polynomial ring R = D[t: a ] . Let

.f (t) = aotn

+ altn-' + . . . + a,.

(1.3.6)

a, E D .

It-e wish to determine the remainder on dividing f (t) on the right by t

-

b.

We have the identity

for a = 1 . 2 , . . .. Multiplying this on the left by a,_, and summing on

1

gives

where N,(b) = (aP1b)...(ab)b, i > 0. No(b) = 1. Hence the remainder on dividing f ( t )on the right by t

Evidently this implies

-

b is

(1.3.9)

16

I . Skew Polyrlomials and Divisiorl Algebras

xt

Proposition 1.3.11. If f ( t )= a,tnpi E D [ t ;a ] and b t D then ( t - b) ai&-,(b) = 0 u~herehJ,(b) is defined by (1.3.9). f ( t ) if and only if

1,-

Now suppose that a pourer o f a is inner and that t h e least such positive power is nr = I,,. W e consider t h e t.s.m. (two-sided maximal) elements o f R . T h e o r e m 1.1.22 gives t h e determination o f t h e two-sided elements o f R. It is clear from this t h a t t h e t.s.nl. elements are t o within units t and t h e central polynomials (1.1.23) which are irreducible i n t h e usual sense i n Cent R . Theorem 1.3.12. Let R = D l t ; a ] where gT for r > 0 is the inner automorph,ism 1%and r is the order of n modulo inner automorphisms. Let c ( t ) = -1 f ?lu-'tr f . . . ysu-"tsr where the y, t Cent D , y ,# 0 an,d euery y,u-% I i i v ( a ) . Suppose c ( t ) is irreducible i n Cent R and there exists a b E D such that

+

n;

Th,en ? ; ' ~ ~ ~ c (=t ) s ( t - b,) where the b, are 5-conjugate t o b. Moreover: R / R c ( t ) -. Afr,(Db) where D b is th,e division subring of elements a such that a a = bab-l.

Proof. T h e condition o n b gives ( t - b) 1 . c ( t ) . Since c ( t ) is t.s.in. it follows from Theorem 1.2.19 t h a t c ( t ) is a product o f factors similar t o t - b. T h e n ? ; l u s c ( t ) has t h e indicated factorization. T h e statement o n t h e structure o f R / R c ( t ) follows also frorn Theorern 1.2.19 and t h e determination o f t h e idealizer o f R(t - b). In t h e special case o f a n ordinary polynomial ring R = D [ t ]w e have r and u = 1. T h e n 1.3.12 specializes t o

=

1

Corollary 1.3.14. Let D be a central diui.sion algebra over F ; f ( t ) a n irred~~cible monic polynomial i n F [ t ] . Assume there exists a b E D such that f ( b ) = 0 . T h e n f ( t ) = I I ( t - bi) i n D [ t ] where the b, are conj7~gatesof b. ( ~ ) ) D ~ ( "is the centralizer of F ( b ) Moreover, D [ t ] / D [ tf ]( t ) -. A ~ , , ( D ~ where in D.

T h e proof is clear. remark that t h e first statement generalizes a classical theorem o f Wedderburiz's [22] and t h e second can b e deduced from a known result o n centralizers i n central simple algebras using t h e fact that D [t]/ D [t]f ( t )-. D @ F F ( b ) . (\Vedderburn's proof o f his theorem will be given i n Chapter 2 (p. 66).) Another special case o f Theorem 1.3.12 is Corollary 1.3.15. Suppose 5 is an automorphism of a division ring D that is of order r modulo inner automorphisms and let 5' = I, where 5u = u. Suppose there exists a b E D such that N,(b) = ( ~ ~ - ~ b ) ( .a. b~=~u.~ Tbh )e n.

1.3. Applications to Skew-polynomial Rings

tr - u

= n I ( t - b,) where the b, are 0-conjugates of b and R/R(tr - u ) 121r(Db) ~ ~ h e D r eo = { a E D I Da = bnb-I).

17

--

Proof. It is clear that u - I t r - 1 is an irreducible element in Cent R . Then the result follows from Theorem 1.3.12. \Vc also have the following generalization of a theorem of Albert's [39, p. 1841.

Theorem 1.3.16. Let a , u , r ,D be as i n 1.3.15 and assume r is a prime. Then R/R(tr - I L ) is a division ring i f and only i f n o b exists i n D such that Nr(b)= u. Proof. tr - u is t.s.m. Hence tr - u is a product of similar factors (1.2.19). Sincc r is a prime and similar polynomials have the same degree either tr - u is irreducible in R or it is a product of linear factors. It is clear also that the second alternative occurs if and only if there exists a b E D such that N,(b) = u . If tr - u is irreducible then R / R ( t r - u ) is a division ring by Theorem 1.2.19. Otherwise, R/R(tr - u) .- &Ir(Db) where N r ( b ) = u. We suppose next that R is a differential polynomial ring D [ t ;61. We shall assume also that char D = p # 0. We recall the followirlg p-power formula in any ring of prime characteristic p:

where i s i ( a , b) is the coefficient of Xi-'

in

+

[. . . [ [ a Xa , b]: Xa+b], ..., X a + b ] , ( p - l ) - ( X a + b ) (See Jacobson. [ 6 2 ] , p. 1 8 7 ) . For p = 2 . 3 . 5 the formulas (1.3.17) are respectively

( a + b)' = a2

+ b2 + [a!b]

+ b3 + [ [ b , a ] , a+] [ [ a ,b],b] , b],b ] ,b] ( a + b)5 = a5 + b5 + [ [ [ [ ab], ( a + b)3 = a3

+ 2[[[[b:aI,bl, b1,aI + 2 [ [ [ [ b , a l , abl, l , bl + 2 [ [ [ ~ b , a l , b l ~ a l+! b2 l[ [ [ [ a , b l , a l , a l , b l + 2 [ [ [ [ abl,, bl, a ] ,a1 + 2 [ [ [ [ ab1,, a ] ,bl, a1 + [ [ [ [ ba ,] ,a ] ,a ] ,a ] .

Since [t,b] = 6b if b E D we obtain

I. Skew Polynomials and Division Algebra5

18

where

V p ( b )= bP + 6 p - 1 b +

.

*

(1.3.20)

where * is a sum of commutators of b, Sb, . . . 6pp2b.For example. for p = 2 . 3 . 5 we have respectively

+ 6b V 3 ( b )= b3 + h2b + [6b,b] V5(b) = b5 + 64b + [[[6b, b ] ,b ] ,b] V2(b) = b2

+ 26[[6b,b],b] + 2[[62b,b ] ,b] + 2[6[6b,b].b] + 2[S3b,b] + 26'[6b, b] + 26[h2b,b].

If D is commutative then

* is 0 and we have the simple formula

We can iterate (1.3.19) to obtain

(t - b y e = t p e - V p e ( b )

(1.3.22)

e

where V p e ( b ) = V p e ( b ) = i/p(Vp. . . ( ~ a ( b ). .) .) As in the twisted polynomial case this leads to the following result: If

+

f ( t )= aotpe altpe-'

+ . . . +aet +d

(1.3.23)

with a, E D then

Hence we have Proposition 1.3.25. If char D = p and f ( t ) E D [ t ;S] has the form (1.3.23) t h e n (t - b) ,1 f ( t ) if and only if

aoVpe( b ) + a l V p e -(~b ) + . . . + aeb

+d = 0.

(1.3.26)

We now assume that R = D [ t ;61 has center F [ z ] as in Theorem 1.1.32 (char D = 6 ) . that is. we have -/, E F = Cent D n Const S and d E Const h such that z = t p e y l t p e - l . . . + y e t - d and 6pe n I 1 ~ p e - l . . . ~~6 = t d . Then we have

+

+

+

+ +

Theorem 1.3.27. Let char D = p and assume R = D [ t ;61 has center F [ z ] as i n Theorem 1.1.32. T h e n R / R z is simple artinian. Moreover, z = n y e ( t- hi) where the b, are 6-conjugate if and only i f there exists a b E D such that d = V p e (b)

+ ?lVp=-l( b ) + . . . + yeb

(1.3.28)

1.4. Cyclic and Generalized Cyclic Algebras

19

and this holds if and only if R / R z -. l\fpe ( D b )where D b is the division subring of D o,f elements a such that 6a = [b,a ] . If e = 1, so z = tP + yt - d then R / R z is a division ring if and only if there exists no b i n D such that d = VP(b)+yb. The proof is similar to that given in the twisted polynomial case and is left to the reader.

1.4. Cyclic and Generalized Cyclic Algebras In this section and the next we shall consider division algebras D that are finite dimensional over their centers and we shall use our constructions to obtain extensions that are finite dimensional over their centers. We begin with a division algebra D that is finite dimensional over its center C and an automorphism a of D such that cr I C is of finite order r . Then, by the SkolernNoether theorem, or is an inner automorphism I,. hloreover. r is the order of a modulo inner automorphisms. We have seen in Theorem 1.1.22 that we can normalize the element u so that a u = u and that if F = C n Inv ( a ) and R = D [ t ;a ] then Cent R is the set of polynomials

Equivalently. Cent R = F [ z ] . z = u - l t r . It is clear that t r - u is t.s.m. and that IL can be replaced by -/uwhere y # 0 is in F . We shall call R / R ( t r - y u ) a generalzzed cyclzc algebra and shall denote it as ( D ,a, y u ) to indicate the ingredients defining it. We have seen that this is simple and we shall see in a moment that it is finite dimensional over F. The classical case is that in which D = C . Then u = 1 and ( C ,a, y ) is called a cyclzc algebra over F. We recall that the dimensionality of a finite dimensional central simple algebra is a square (BA 11. p. 222). Now suppose [D : C ] = n2.We have [ C : F] = r by Galois theory. Using the division process we see that any element of A = ( D ,u, u) has a representative of the form ao a l t . . . a T p l t T - I where a, E D and ao a l t + . . . + a,-ltr-l 6 R ( t r - u ) if and only if every a , = 0. Now A contains the subring D in the obvious way and we have the left dimensionality [ A : Dlt = r . Hence

+

+

[ A :F] = [ A : Dje[D : C ] [ C F] : = n2r2

+ +

(1.4.2)

and in the special case in which D = C we have

Proposition 1.4.4. C is the centralizer of D in A and F is the center of A.

a(ao

+ a l t +. . . + a,-ltr-l)

r (a0

+ a l t + . . . + a,-ltr-')a

(mod R ( t r - y ~ ) )

I. Skew Polynomials and Division Algebras

20

<

then aa, = a , ( a 2 a ) , 0 z 5 r - I. Since r is the order of a modulo inner automorphisms the foregoing commutativity holds for all a E D if and only if every a, = 0 for i > 0 and a0 E C . This proves the first staternent. The second follows by considering conlrnutativity with t . We now see that A is central simple over F with [A : F ] = n 2 r 2 and in the cyclic case A = ( C ,a , ? ) , [ A : F ] = r 2 . The standard argument used in the case of polynomials in one indeterminate over a field shows that A is a division algebra if and only if tT - y u is irreducible in D [ t ;a ] (see e.g. B A I , p. 131). In general, t T - u = plp2.. . p , where the p, are irreducible and similar and hence have the same degree m (Theorem 1.2.19). Then r = m s and A -. hf,(A) where A -. B,/Rp,. B , the idealizer of Rp,. Then s 2 [ A : F ] = [ A : F ] = n 2 r 2 = n 2 m 2 s 2gives [ A : F] = n2m2and the degree (= square root of dimensionality) of A over F is nm = n deg p,. We recall also that s = r and m = 1 if and only if there exists a b E D such that N,(b) = ( ~ ~ ~ ' b ) ( a. .~. b- = ~ u. b )Then A is isomorphic to the subring D b of D elements a such that a a = hub-'. Then [ A: F] = [ D : C ] = n 2 . It is readily seen using shortest relations that elements of D b that are F-independent are C-independent. Hence [ C D b : C ] = [ D b : F ] = n2 so C D b = D . Then D -. C % F D Y~ C ~ F A . If we put x = t R ( t r - 2 u ) then ( 1 .x , . . . x T P 1 )is a base for A as vector space over D and we have the relations

+

.

Conversely, suppose A is an algebra containing D as a subalgebra and containing an element x such that (1.4.5) holds where D , a and u satisfy the conditions stated at the beginning of this section. Then we have a homomorphisin 7 of R = D [ t ;a ] into A such that

Evidently ker 77 > R ( t r - y u ) and since R ( t r - y u ) is a maximal ideal ker 17 = R ( t T - ~ u )Hence . the subalgebra of A generated by D and x is isomorphic to R / R ( t T - r u ) and if A itself is generated by D and z then A is isomorphic to the generalized cyclic algebra ( D ,a , y u ) . Generalized cyclic extensions are analogous to simple algebraic extensions of fields. We shall now construct analogues of simple transcelldental extensions by central localization.

Theorem 1.4.6. Let R = D [ t ;a ] where D is finite dimensional over its center C , a / C is of finite order r: a" = I,, a u = u . Then the localization R s for S the monoid of n,on-zero elements of Cent R = F [ z ] z, = u-It", is a division ring whose center is the ,field of fraction,s F ( z ) of F [ z ] .Moreover: the can,onical m o p of R into R s is injective.

+

+

Proof. The elements of R s have the form f ( t ) /y ( z ) where p ( z ) = yo ylz . . y,zm, y, E F . Since R is a domain it is clear that the canonical map

+

1.5. Generalized Differential Extensions

21

f ( t ) -i f ( t ) / l of R into Rs is a monomorphism and that the center of Rs is F ( a ) , the set of elements d ( z ) / p ( z ) $. ( z ) ,p ( a ) E F [ z ] .We can identify R with the corresponding iubrirlg of Rs.To see that Rs is a division ring it suffices to show that every f ( t ) # 0 in R is invertible in Rs.This will follow by showing that for every f ( t ) # 0 there exists a cp(z) # 0 in Cent R such that f ( t ) is a right and a left factor of p ( z ) . To see this we note that since [D : F] = [D : C] [ C : F ] = n 2 r , the dimerlsionality of the vector space over F of polynomials g ( t ) with deg g < deg f is n 2 r deg f . We now divide z Z ,0 5 7 5 n 2 r deg f by f ( t ) obtaining

+

where deg g,(t) < deg f ( t ) .Since the number of g, is n 2 r deg f 1 there exist r , E F not all 0 such that Cy,y, = 0. Then p ( a ) = Cy,z" 0 and f ( t ),1 y ( a ) . Since cp(z) E Cent R and R is a domain we also have f ( t ) It p ( a ) . We now identify R and F ( a ) with their images in Rs and we put F = F ( z ) , R = Rs = FR,D = FD, c = FC. Then since [D : F] < no, [D : F ] < cc and since D is a domain. D is a division ring. It is readily seen also that the center of D is C . The inner automorphism It of R stabilizes D and 3 = It D is an autorrlorphism of D whose restriction to 6 has order r . Moreover, 5 T = I% = I U Z .We have tZ = ( 5 6 ) t and t r = 7 ~ 2 It . follows that R -. ( D .5 ,u z ) . VLTenow specialize to the case in which D = C. Then C is a cyclic field over F and c is a generator of Gal C I F . The center of R = C[t;~j is F [ a ]a, = t r and R is the cyclic division algebra ( 6 , 6 ,z ) over F = F ( a ) .

1.5. Generalized Differential Extensions Suppose first that D = C is a field of characteristic p # 0 , 6 is a derivation in C . F the subfield of 6-constants and R = C [ t ;61.Let C [ 6 ]be the subring of End C generated by the multiplications x --, ex in C and the derivation 6. We have the canonical homomorphism u of R onto C[6] fixing the elements of C and mapping t into 6.

Lemma 1.5.1. Either ker u = 0, so v is an isomorphism, or there exists a p-polynomial f ( A ) = Ape y l ~ p e - l + . . . y,A with coefficients i n F such that f ( 6 ) = 0 . In the latter case, if f ( A ) is chosen with minimal e then ker u = Rz where a = f ( t ) = t p e yltpe-l . . . -iet hforeover, [C[6] : CIe = pe and f ( A ) is the minimum polynom,ial of S as a linear transformation of C as ?lector space over F .

+

+

+

+ +

Proof. We know that ker u = Rw* where w* is a two-sided element of R.By Thcorem 1.1.32,we may assume xi* E Cent R,and Cent R = F unless we have a p-polynomial f ( A ) E F [ A ]such that f ( 6 ) = 0. Since ker u # R the first case implies that ker u = 0. In the second case if f ( A ) = Ape . . . is chosen

+

22

I. Skew Polynomials and Division Algebras

with e minimal then Cent R = F[z] where z = f ( t ) . On the other hand, z E ker u since f (6) = 0. Hence ker u = R z . The map u is also a C-module homoniorphism of R onto C[6] and hence C[6] and R I R z are isomorphic Cmodules and hence [C[6] : C]! = [RIRz : C]< = deg z = pe. Also z is the polynomial of least degree contained in Rz, hence, the polynomial of least degree wltll coefficients in C such that bo b16 +. . . = 0. Since the coefficients of z are in F it is clear that j(A) is the minimum polynomial of 6 as linear transforrrlation in C over F .

+

In the first case of the lemma we say that 6 is transcendental and in the second 6 is algebrazc and the degree pe of its minimum polynomial is called the de,qree of 6.

Lemma 1.5.2. If 6 is algebraic of degree pe then [ C : F] = pe and C is purely inseparable of exponent one over F. Moreover? C[6] = EndPC. Proof. This is an immediate consequence of the Jacobson-Bourbaki theorem (BA 11. p. 471). For. C[6] is a ring of endomorphisms of C containing the set of multiplications (that can be identified with C ) and [C[6] : CIe = pe. The subfield of F corresponding to C[6] in the Jacobson-Bourbaki correspondence is the set of c E C such that c6 = Sc. This is F. Hence [ C : F ] = pe and C[6] = EndPC. The fact that C is purely inseparable of exponent one over F is clear since for any c E C, 6(cp) = pcP-l(S~)= 0 SO c p E F . Evidently the subfield F of C is 1-dimensional over F and since 6 is an F-linear transformation and F = ker 6, we have the useful fact that 6 C is a hyperplane in C I F . that is. [6C : F] = pe - 1. We now suppose that D is a division ring that is finite dimensional over its center C of characteristic p and 6 is a derivation in D such that 6 1 C is algebraic with minimum polynomial f (A) of degree pe. Then f (6) is a derivation in D such that f (6)C = 0 by 1.5.1. Thus f (6) is a derivation in D over C and since D is finite dimensional central simple over C it follows from the derivation analogue of the Skolem-Noether theorem that f (6) is an inner derivation (see exs. 10. 11. p. 226 of BA 11). We now have

Lemma 1.5.3. The element d E D such thut f (6) = id can be chosen so that Sd = 0. Proof. Let d be any element of D such that f (6) = zd. We have z6d = [6,zd] = [6. f (6)] = 0. Hence 6d = c E C . Put g(X) = A-I f (A) E FIX]. Then g(6)c = f ( 6 ) d = 0 so c E V = {x E C / g(6)x = 0). Kow V is an F-subspace of C containing 6C and. V # C since deg g < deg f and f (A) is the minimum polynomial of the linear transformation 6 of C I F . Since [SC : F] = pe - 1 it follows that V = 6C. Hence c E 6C and we have a c' E C such that 6c' = c. Replacing d by d - c' gives a d satisfying the required condition. We now put z = f (t) - d where d is as in 1.5.3. Then the center of D[t: 61 is F [ z ] .We denote the algebra RIRz where R = D[t; 61 by A = (D, 6, d) and call

1.5. Generalized Differential Extensions

23

this a generalized differential extension o f D . These extensions are analogous t o t h e generalized cyclic extensiorls we considered i n t h e last section. In t h e special case i n which D = C : we have t h e differential extension A = ( C ,6: d ) . W e now have Theorem 1.5.4. Let D be finite dimensional over its center C of characteristic p and let 6 be a derivation i n D such that 6 1 C is algebraic with . . . y,A. Choose d E D such m i n i m u m polynomial f ( A ) = Ape ylApe-' that i d = f ( 6 ) and 6d = 0 . Let A be the generalized differential extension R / R z = ( D !6,d ) where R = D [ t ;61, z = f ( t )- d . T h e n A is central simple over F = C n Const S and [ A: F ] = p2en2where [ D : C ] = 71'.

+

+ +

Proof. Since t h e center o f R is F [ z ]b y Theorem 1.1.32, i t is clear t h a t z is a t.s.m. element o f R . Hence A = R / R z is simple. T h e cosets o f R z i n R / R z have unique representations o f t h e form ~g~~~ a i t Z , ai E D , and D can b e identified w i t h its image i n R. Now suppose Ca,ti R z is i n t h e center o f A. T h e n [ C a i t Za] , for a E D and [ C a i t i ,t] are divisible b y z . Hence [Gait" a ] = 0 = [ C a Z t zt ,] so C a i t i E Cent R and b y Theorem 1.1.33, C a i t Z = 7 E F . T h u s F = Cent A and A is central simple over F . W e have [ A: Dle = p e , [ D : C ] = n2 and [ C : F ] = p" ( b y 1.5.2). Hence [ A: F ] = p Z e n 2 .

+

\lie remark t h a t t h e argument used t o show that t h e center o f A is F shows also t h a t t h e centralizer o f D i n A is t h e center C o f D . Now put x = t R . T h e n ( 1 ,z , . . . . ape-') is a base for A as vector space over D and we have t h e defining relations

+

xa = ax

+ Sa, x p e + y l x P + . . . + 7, = d. e-1

(1.5.5)

4 s i n t h e generalized cyclic case, these characterize t h e algebras ( D ,6, d ) . W e also have t h e following theorem which is analogous t o Theorem 1.4.6. Theorem 1.5.6. Let D , 6, R, C : F , d , f ( A ) ,z be as i n Theorem 1.5.4. T h e n the localization Rs for S the monoid of non-zero elemen,ts of Cent R = F [ z ] is a division ring whose center is the field of fractions F ( z ) of F [ z ] . The canonical m a p of R into Rs is injective.

W e omit t h e proof which is similar t o that o f 1 4.6. Also as i n t h e discussion o f t h e generalized cyclic case, i f we put F = F ( z ) , R = Rs = FR. D = FD. c = FC t h e n D is a division ring w i t h center 2' and t h e inner derivation zt o f R stab~lizesD.Let 8 = zt I D. T h e n 6 1 C? is algebraic w i t h m i n i m u m polynomial f ( A ) . I t is readily seen that R = ( D ,6,d + z ) . We omit t h e details. Xow suppose D = C . T h e n t h e element d can b e chosen t o b e arly element i n F . Theri C is a rnaxirnal subfield o f t h e differential extension ( C ,6 , ~ )T.h e center o f R = C [ t ;61 is F [ z ]where z = t p e yltpe-' .; . +ye - y . In particular 6 , z ) and this is a central we can choose y = 0. T h e localization R = Rs is (c. division algebra over F ( z ).

+

+

24

I . Skew Polynomials and Division Algebras

1.6. Reduced Characteristic Polynomial, Trace and Norm In this section we shall give a definition of the rcduccd (or generic) characteristic polynomial, trace arid norm function of a finite dimensional associative algebra, and derive some properties of these functions for central simple algebras. We shall use the reduced characteristic polynomial to obtain quick and natural proofs of the existence of separable splitting fields of central simple algebras and of the existence of a single generator of a separable commutative algebra over an infinite field. Some of these results will be used in the next section and others in Chapter 2. Let A be a finite dimensional associative algebra over a field F. ( u l , uz. . . . , u,) a base for A over F . If a E A we denote the minimum polynomial of a in A by pa(A). We recall some well known results on p,(A) in the special case in which A = M,(F) (so n = m2). In this case we have the characteristic polynomial X, (A) = det (A1 - a ) and the Hamilton-Cayley theorem that x,(o) = 0. It follows that pa(A) I x,(A). We recall also that mTecan diagonalize the matrix A 1 - a in .Wm(F[A]),that is. we can find invertible matrices P(A). Q(A) E IW~,(F[A])such that where the d, (A) are monic polynomials and d, (A) / d, (A) if i 5 j . Then X, (A) = cl, (A) and we have the sharpening of the Hamilton-Cayley theorem due to Frobenius: &(A) = pa(A). Evidently this implies that p,(A) and xa(A) have the same irreducible factors and the same roots in the algebraic closure F of F. Finally, we have the following formula for pa (A):

ny

-

where A,-l(A) is the g.c.d. of the (m - 1) rowed minors of (A1 - a ) (see BAI. 11. 201). We consider an arbitrary A again with the base (ul, ua.. . . , u,). We introduce n indeterminates &, t2,.. . , and the field F(E) F(C1,(2,. . . , <), of rational expressions in the <,with coefficients in F. Consider AF(,t) = F ( < ) Z F A obtained by extending the base field F of A to F(<). Put a = C[,u, E A = AF(,t).We call this a generzc element of A. We now denote the minimum polynomial of x by m,(A) (rather than p,(A)) and write this as m,(A) = Am - T ~ ( [ ) x ~ -. .~. ( - l ) m ~ m ( < ) (1.6.2)

<,

where

T,

-

+ +

(6) E F ( < ) . 1% now have

Lemma 1.6.3. 7n, (A) E F [ A ! <] F[X,tl,. . . , homogeneous polynomial of degree i in the ['s.

.I,<

More precisely, ri([) ,is a

Proof. Since A has faithful representations by matrices, e.g. a regular representation. we may assume that A is a subalgebra of a matrix algebra M N ( F ) .

1.6. Reduced Characteristic Polynomial, Trace and Norm

25

Then AF(<) 1s a subalgebra of Al,v(F(())and x = CE,u, is a matrix whose entrles are honlogerleous linear expresrions in the [ ' s . Hence the characteristic polynomial x,(A) has the form

mhere t , ( [ )is a homogeneous polynoniial of degree i in the ['s and AT is the degree of the representation. Since x N - t l ( [ ) z N - I . . ( - l ) N t ~ ( (= ) l0 . by the Hamilton-Cayley theorem, and rn, ( A ) is the rninimurn polyrlonlial of I . ( A ) x,(X). It follows from Gauss' lemma that rr~,(A) E FIX, 61 (p. 154 of BAI). Lloreover. since ,y,(A) is homogeneous of degree N in X and the 6's; na,(A) is homogeneous of dcgrcc m in X and t,he E's. It follows that r,(E) in Ci (1.6.2) is homogeneous of degree i in the 6's.

+. +

\Ye r 1 o ~consider ~ ArrCi= F[El,. . . . En] @ F A which is an F-subalgebra of AF(<).Let a = Ca,u, E A. SVe have a n F-algebra homomorphisrrl rl, of F [ [ ] onto F slicli that 6, --i a,. Then we have the homomorphism q, = vu @ 1 of ArIEl onto A(= F X F A). This maps the generic element x = CE,u, into a = Ca,u, arid an) polynomial f (El.. . ,En) into f ( a l , .. . . a,,). Now since

applying 7, we obtain

m,(a) = 0 where

mU(X)= A'"

+ ... + (-l)m~,L(o).

-T ~ ( ~ ) X ' " - ~

(1.6.6) (1.6.7)

It is readily seer1 that nz,(A) is independent of the choice of the base . ; u,) of A I F : Let (vl:v z ? .. . . v,) be a second base for A / F and v, = C,!?i,u,, ( B Z j ) invertible in ll.f,(F). Then Ctivi = C<;u3where 6; = CEi,Oljand mcl,, ( A ) is obtained by replacing [:,by 6;. 1 5 j 5 n, in m,(A). Nor\- if a = Caivi then a = Ca;uj where a: = Ca,P,,. It follows that m,(X) is lmchanged in passing from the base ( u l ; .. . , u,,) to the base ( u l : . . ,!u,). TVe shall call m,(A) the reduced characteristic polynomial of the element a and we call ( 1 1 ~ .uz, . .

t ( a ) = 7-1 ( a ) :n(a)= r m ( a )

(1.6.8)

the reduced trace and reduced norm respectively of a. The integer m is called the degree of the algebra A. \Ire shall see in a moment that if A is central sinlple this coillcides witll the degree of A as previously defined. Let E be a n cxtensiorl field of F and consider the algebra AE = E X F A. \Ve can regard A as contained in AE. Then the base ( I L ~ . . . . . I L , ) of A is also a base for AE. It follows that the reduced trace and reduced norm functions on AE are extensions of the5e functions for A. This rerrlark implies that in developing the properties of these functions there is no loss in generality in assuming the base field is infinite or even algebraically closed. The advantage of working with infinite base fields is that we can use the Zariski topology of the vector space A I F .

I. Skew Polynomials and Division Algebras

26

n7e define the degree of a E A to be the degree of the minimum polynomial pa(A) of a. Proposition 1.6.9. If F is infinite then the elements of A whose degrees are th,e degree of A constitute a non-vacuous Zariski open subset of A.

Proof. If x = Cy Ezuz as before then

where p,,([) 1s a llomogeneous polynomial of degree 1 in the 6's. Since the degree of the minimum polynomial m,(X) of x in = AF(0 is m, the elements x k . O i k < m - 1. are linearly independent in A.Hence there exists a nonzero m-rowed minor in the m x n rnatrix ( p I i z ( [ )Let ) . these non-zero m-rowed . . . , D,([). Then it is clear that the set of elements a = Ca,u, minors be Dl such that deg a = m is the union of the sets defined by D, ( a )# 0.Evidently this is a non-vacuous Zariski open subset of A.

A

(c),

We again consider A as imbedded in A f N ( F ) . \ire have the three polvnomials pa ( A ) .m, ( A ) and the characteristic polynomial X , ( A ) of the matrix ~ ( a )Since . m , ( a ) = 0, p,(A) / m,(A). We also have m,(X) x,(A) since m , ( A ) xs( A ) . Also p,(A) is the rninimum polynomial of the matrix a so p a ( A ) and x a ( A ) have the same irreducible factors. Hence ,u,(A) and m,(X) have the same irreducible factors. We can use this remark to prove Proposition 1.6.11. (i) a is invertible if and only if n ( a ) # 0. (ii) a is nilpotent if and only if m,(A) = Am.

+

Proof. (i) a is invertible if and only if X pa(A) and hence if and only if A m,,(A). Since A m,(A) ++n,(a)= 0:the result is clear. (ii) a is rli1pot)ent if and only if p,(X) = AT for sorne r : hence: if and orlly if m,(A) = Am.

+

It is clear from 1.6.11 (i) that if F is infinite then the invertible elements of A form a non-vacuous Zariski open subset. By (1.6.6) and (1.6.7) we have

We now define the reduced adjoint a# of a by

Then we have

aa# = n ( a )1 = a#a.

(1.6.13)

Now suppose A is central simple over F. Assume first that A = h17,(F) and let {e,, / 1 5 2 . 2 5 m ) be the usual matrix base for A f m ( F ) . Then we have the generic element X = (&). The characteristic polynomial x x ( A ) is

1.6. Reduced Characteristic Polvnomial, Trace and Norm

27

irreducible in F[E,A] since this is homogeneous of degree n in the ('s and A and ~ ~ (= 0 det ) X is irreducible in F [ ( ] .It follows that ,yx(A) is irreducible in F (6)[ A ] a.nd hence t,hat mx ( A ) = ,yx ( A ) .Then for any a E ( F ), m , ( A ) = x a ( A ) = det(X1 - a ) . Now let A be any cesitral simple algebra and let E be a splitting field for A (e.g. E = F the algebraic closure of F ) . Then -AE = M m ( E ) . Regarding a as contained in iZ'I,,(E) we have ma(A) = det(A1 - a ) , since m,,(A) is unchanged on extension of the base field. Thus det(X1- a ) E F [ A ] .hloreover, the degree of A is .rn which is the same as the degree of A as defined earlier, since [ A: F ] = [AE: El = m 2 . It is clear from the properties of matrices that

*

n(cu1) = am if a E F ,

n(ab) = n ( a ) n ( b )

t ( 1 ) = m, t ( a b ) = t ( b a ) .

(1.6.14) (1.6.15)

These results hold for arbitrary A. We shall derive these and other properties of the generic norm and trace in a rnore general situation in Chapter V (see Sec. 5 . 3 ancl 5.5). Again let A = ,Wm(F),X = (t,,). Let d ( X ) be the discriminant of nl,x( A ) = ,yx ( A ) .Now d ( X ) # 0 since the ['s can be specialized in F or in an infinite extension field of F to give a matrix with distinct characteristic roots. Now let A be a central simple algebra of degree rn over F , ( u r J ) :1 5 i, j 5 r r ~ ) a base for A, x = C t i j u t j . Then the discriminant d ( x ) of m,(A) is not zero since this is obtained by a linear change of indeterminates frorn m x ( A ) after passing to AE = hfT,(E) for a suitable extension field E of F . Now the set of a such that ' m a ( A )has distinct roots (in the algebraic closure of F ) is the set defined by d ( a ) # 0. Hence if F is infinite this set is a non-vacuous Zariski open subset of A. hloreover, for these elements we have m a ( A ) = pa(A). We can use this result to give a simple proof of the existence of separable splitting fields for central simple algebras. First we prove Theorem 1.6.16. Let D be a central division algebra of degree m over an

infinite field F . Then the subset S of D of elements a such that F ( a ) is n .sepa,rable subfield of din~ensionalityrn over F is non-vacuous Zariski open. Proof. It is clear that S is the set of a such that m,(A) has distinct roots and we have seen that this is non-vacuous Zariski open. We recall that if A is central simple of degree m and E is a subfield of A such that [E : F] = m then E is a splitting field. (BA 11, p. 224). 1%shall slow give a proof of this tlleorern which is due to Wedderburn [21]and which gives additional information on the reduced characteristic polynomial. Theorem 1.6.17 (Wedderburn [21]).Let A be central simple of degree m, E a subfield of A such that [ E : F ] = m. Then E is a splitting field for A .

Proof. We regard A as (left) vector space over E. Then m2 = [A : F ] = [ A : Elt [ E : F ] and hence [ A : Ele = m . Let ( u l ,v2, . . . , urn) be a base for A over

28

I. Skew Polynomials and Division Algebras

E. 1% obtain a representation p of A by matrices in hf7,(E) by writing m

p,, (a)v,, i 5 i 5 rn.

v,a =

(1.6.18)

,=I

Then a --i p(a) is a rnonornorpliism of A / F into h f m ( E ) / F and we can identify A with its image. Since A is central si~npleover F, Af,,,(E) -. A 8~C where C is the centralizer of A in 111,(E) (BA 11, p. 218). Now, C contains E and since [Afm(E) : F] = rn3 = [A : F ] [ C: F]. so C = E. T h m A 8~ E -. iZ/I,(E). We now observe that, since the reduced characteristic polyr~omialis unchanged under extension of the base field. if a E A, then m,(X) = x,(,)(X). Thus the characteristic polynomial of p(a) is contained in F[X].In particular. n ( a ) = det p(a) E F . We can now prove the following theorem which was first proved by ICiithe 1331. Theorem 1.6.19. A n y finite dimer~sionalcentral simple algebra has separable splitting fields.

Proof. If A is finite dimensional central simple A = hI,(D) where D is a central division algebra. Hence it sliffices to prove that D has a separable splitting field. If F is finite then D = F by Wedderburn's theorem on the commutativity of finite division rings. Hence we may assume F infinite. Tlieli D has separable subfields E whose dimensionalities are the degree of D by 1.6.16. Such a subfield is a splitting field for D by 1.6.17. Then this is a splitting field for A. We recall that an algebra A is called separable if AF is semi-simple for F the algebraic closure of F. A commutative algebra A is separable if and only if AF = F e l @ . . . @ F e n where the e, are non-zero idempotents that are orthogonal (e,e, = 0,i # 3 ) . It follows that F[a] is separable if and only if the minimum polynomial p,(X) of a has distinct roots. The argument used to prove 1.6.16 gives the following extension of that result: Theorem 1.6.20. Let A be central simple of degree n over a n infinite field F. T h e n the subset S of elements s u ~ hthat F[a] is separable and [F[n] : F] = n 2,s non-ua,cuous Zariski open.

We also have the following extension of the classical theorem on the existence of primitive elements for separa,ble field extensions (with infinite base field). Theorem 1.6.21. If A is separable and commutati~ieover a n infinite field F t h e n there exist a E A such that A = F [ a ] .

1.7. Norm Conditions

29

Proof. We have AF = Pel @ . . . 8 F e n where the ei are non-zero orthogonal idempotents. If the Ci are indeterminates and z = CEzezthen m,(A) = 17(X t , ) and d(x) # 0 for the discriminant d ( z ) of m,(A). It follows that the set S of a E A for which m,(A) has distinct roots is non-vacuous Zariski open. For these a : the minimum polynomial p,(A) = m, (A) and since [F[a]: F] = n, F [ a ] = A. We remark that if F is finite then the conclusion of 1.6.21 may fail. For example; lct A = A1 3 . . . 8 A, where IF = pk and Ai is a field such that IAil = p " . I f A = F [ a ] a n d a = a l + . . . + a , wherea, E A, thenwemust have pa% (A) # paJ(A) for i # j and pa%(A) the (irreducible) minimum polynomial of a, in A,. We have deg pa, (A) = !and it is well-known that the number of monic irreducible polynomials of degree !over F is N ( ! , ~ ~ = ) $ Ed,! p(!/d)pkd ( p here is the h?iibius function). It follows that if s > N(!, pk) then there exists no a such that F [ a ] = A. The argument shows also that if s 5 N ( k , p k )then we do have a single generator and it is easy to extend it to determine a condition that any finite commutative separable algebra have a single generat,or.

1.7. Norm Conditions fire shall now apply some of the results of the last section to the study of the polynomial rings C[t;a] and C[t;S] where in the first case C is a cyclic field with [C : F] = r and a is a generator of Gal C I F and in the second case C is purely inseparable of exponent one over F and S is a derivation in C which is algebraic with Const 6 = F. MTeconsider first the case R = C[t;a ] . This can be imbedded in the division ring R = Rs where S is the set of non-zero elements of the center F[z], z = t r , of R. We have seen in Section 1.4 that R is thc cyclic algebra (C, 5,z) over F = F ( z ) where 6 = FC and 5 1 C = a. Let n denote the reduced norm on ( 6 . 6 ,z) over F . Then we have Proposition 1.7.1. (i) I f f ( t ) E R then n(f (t)) E F[z] and f (t) n(f (t)) in

R.

(7;i) If

f (t) = a0 + a l t + . . . + a,tm

where the a, E C and m


(1.7.2)

then

+ + (-~)'"~VC~F(~,)Z~.

n ( f (t)) = ATcl~(ao) . . .

(1.7.3)

Proof. (i) we have [C : F] = r. [R : p] = r 2 and (1, t , . . . , t r - l ) is a base for R over C. Jlle can use the matrix p ( f ( t ) ) as in (1.6.18) to compute n(f ( t ) ) . Since t r = z. f ( t ) can be written as a linear combination of ( l , t , .. . . t i ' ) with coefficients in C[z]. Hence the matrix p(f ( t ) ) has entries in C[z] so n ( f(t)) = det p(f (t)) E C[z] n F = F[z]. In a similar manner one sees that

30

I. Skew Polynomials and Division Algebras

all the coefficients of ~ ~ ( f ( (A) ~ ) are ) contained in F [ z ]and hence, by (1.6.12). j ( t ) " E R. Since f ( t )f ( t ) # = n ( f( t ) )= f ( t ) # f ( t ) .f ( t ) 1 n ( f( t ) ) in R. (ii) JVe now assume f ( t ) is as in (1.7.2) with rn < r . Then the matrix

0

u~-nLn,,z

0

ar-m-l

0

0

a.

ar-m-l

al

ur-mao

.

a~-m-l

.

u ~ - mam- I

am

Hence n ( f ( t ) )= det p(f ( t ) )is as indicated in (1.7.3). \Ve recall that if A = 1V/I,,(D)where D is a central division algebra then the degree of D is called the index of A. We have shown in Section 1.4 (p. 20) that if the irreducible factors of f r - y.7 # 0 in F , are of degree m then the index of ( C .0 - 7 )is 777. The following theorem gives another determination of this index.

Theorem 1.7.4 (Jacobson [34]). The index of fhe cyclic algebra ( C ,a , T/) is tlze smallest positive integer m such that (7 - t r ) m= n ( f ) for f E R = C [ t ;01. P'roof. Suppose (y - z)m = n ( f ) where rn is minimal. Then m 5 r since n(-/ - z) = (y - z)'. Now f is irreducible. For, if f = fl f2 where deg fl> 0 then (7- z ) ~ = ' n ( f ) = n ( f l ) n ( f 2 ) Since . f i / n ( f , ) : deg n ( f i ) > 0 and since n ( f , ) E F [ z ] .n ( f , ) = ( y with m, < m , contrary to the minimality of nz. Hence f is irreducible. Since f n ( f ) ;f / ( y - z ) ~ Let . h be the smallest positive integer such t>hatf lc ( y - z ) ~We . cla,irn h = 1. Otherwise: .f Jk ( y - z ) and since f is irreducible, ( f ,y - z)e = 1 and hence, b y (1.3.2), deg[f,y - z ] , = deg f deg(7 - z ) . It follows that (? - z ) f = f ( y - z ) = [f.? - z], and since f le (7- z)", ( y - z) f ( y - z)lL.Then f lc (7- z ) ~ - ' contrary to the choice of h. Thus f is an irreducible left factor of 7 - z so r . By (1.7.3) and n ( y - z ) = (7- z)" we see that deg n(f ) = r deg f . deg f Hence r m = deg(y - 2)"" deg n ( f )= r deg f . Then deg f = m and hence m is the index of (C:a, y ) .

+

<

An immediate consequence of Theorem 1.7.4 is Wedderburn's .'norm condition" :

Corollary 1.7.5 (Wedderburn [14]). If q is the least positive integer such that 7 4 = N C I F ( B )h; E C , th,erl the index of (C,u , y) is q.

>

Proof. By 1.7.4, if m is the index then (7- z ) =~n ( f ) for f = b by 1.7.3. y = NclF(b). Hence m q.

>

+ . ... Then,

1.8.Derivations of Purely Inseparable Extensions of Exponent One

31

Now suppose that C is a field of characteristic p: 6 an algebraic derivaT 1 ~ ~ e - l ... yeX E F [ X ] , tion in C with minimum p-polynomial X p e F = Const 6. Let R = C [ t ;61.Then Cent R = F [ z ] , z = tpe+-yltpe-I+.. .+yet. Let R = Rs where S is the monoid of non-zero elements of F [ z ] .Then R is a central division algebra over F = F ( z ) and R is the differential extension ( 6 , 6:z ) where c = FC and / C = 6. We have the reduced norm on R mapping R into F and this maps R into F[z]. Moreover, as in 1.7.1 f ( t ) n(f ( t ) ) and if f ( t ) = a0 alt . . . amtm where the at E C and m < r then LV(f( t ) )= *a
+

+

+

S

+ + +

+

Theorem 1.7.6. The index of the differential extension ( C ,6 , ~ is) the smallest power pY such that (? - z)pq = n ( f ) for some f E R. We omit the details.

1.8. Derivations of Purely Inseparable Extensions of

Exponent One Let CIF be a finite dimensional purely inseparable field of exponent one and characteristic p. Then C = F ( u 1 . . . . , u,) F ( I I I 8~ ) F ( u z ) X F . . . % F F(u,) where u3)= a, E F and [ F ( u l ): F ] = p, so [C : F ] = pTn (see BAII, p. 541). In this section we shall derive some results on derivations of C tliat will be needed for the study of p-algebras (Chapter 4). The first of these is a classical result due to Baer [27]:

Theorem 1.8.1. There exzsts a derzvatzon 6 of C / F whose field of constanfs. Const F = F. Proof. Using inductioli on m it suffices to show that if E is a subfield of C I F and 6 is a derivation in E I F such that Const 6 = F then 6 can be extended to a derivation 6' of any E ( u ) such that up = a E F and [ E ( u ): E] = p without increasing the field of constants. Now 61 = 0. Since E / F is finite dimensional it follows that 6 is not surjective on E . Thus there exists a b E E . @ 6 E . It is readily seen that if e is any element of E then there exists a unique extension of 6 to a derivation 6' of E ( u ) such that 6'u = e. Let 6' be the extension so that 6'u = b. We claim that Const 6' = F . Let c E Const 6' and write c = a0 a l u . . . a,-lup-l, a, E E , in terms of the base (1.u , . . . . u p p 1 )of E ( u ) / E . Then 6'c = 0 implies the following system of equations:

+

+ +

32

I. Skew Polynomials and Division Algebras

+

The last of tliese implies that a,-1 E F. Then 6ap-a (p - l)a,-lb irnplies a,-l = 0 since a,_l # 0 in F gives b = ( ( p - l)a,-l)-1Sa,-2 = S((pl)a,-l)-1a,-2) contradicts b $ 6E. Continuing in this may we obtain a, = 0. L = I . . . . . p - 1. Then c = no t E and 0 = S'c = bc implies that c E F . Now suppose A I F is finite dimensional simple with center C. lye shall show that any derivation 6 of C I F can be extended to a derivation of A. lVe pro\-e first the following

Lemma 1.8.2. Let A be a ,finite dimensional algebra over field of F . T h e n Der A E / E -. (Der A / F ) E .

F ;E a n extension

Proof. First, it is clear that if 6 E Der A t,he extension of 5 to a linear be a base transformation of A E / E is a derivation of A E I E . Let (211: . . . :u,,) for A / F and hence for A E / E . If 6 E EndpA then 6 E Der A / F if and only if S(u,u,) = .u,(6.n3) + (6,ui)uJ for 1 i ; j n. These corlditiorls are equivalent s equations with coefficients in F . The to a system of n2 h o ~ n o ~ e n e o ulinear maximum number of linearly independent solutions in F of this system is the same as its number in E . It follows that the extension to AE of a base for Der A I F is a base for Der A E / E . This irrlplies t,hat Der A E I E -v (Der A / F ) E . 17

<

<

As before, let A be finite dim~nsionalo w r F with center C a purely inseparable extension of exponent 1 over F. If 6 is a derivation of A I F , then 6 maps C into itself and hence we have the homomorpllisni rj : 6 --i 6 1 C of Der A I F into Der C l F. By the derivation analogue of the Skolem-Soether theorem (exs. 10. 11, p. 226 of BA 11) Ker 17 is the subspace Inder A of inner derivations. Inder A ii Der A +17 Der C . Hence we have the exact sequence 0

Theorem 1.8.3 (Jacobson [371],p. 220). Let A be finite dimensional simple ouer F 111ithcenter C a purely in,separable ,field o,f exponent one over F . T h e n

0

+ Inder

A

-+

Der A 3 Der C

+0

(1.8.4)

Proof. Suppose first that A = 1bIn,(C) and let 6' E D e r ~ c Then . (a,,) --t (S'a,,) is a derivation 6 of I % I ~ ( C ) /such F that 6 1 C = 6'. Hence (1.8.4) is exact. Now suppose A is finite dimensional simple over F with center C purely inscparablc of cxponent one over F . By Theorem 1.6.19, A I C has a finite dimensional separable splitting field E I C . so E zc A -. hlrn(E). Now E is a separable extension of the purely inseparable field C / F and hence E -. S ~ C F where S is the maximal separable subfield of E/F (BA 11, p. 492). Then

1.9. Some Tensor Product Corlstructioils

33

Hence. by 1.8.2: [Der A : F] = [Der As : S] = [Der Al,,(E) : S]

(1.8.5)

where 121,,(E) is regarded as an dlgebra over S . Now E is purely Inseparable of exponent one over S . Hence the exactness of (1.8.4) holds for .lIm(E) as algebra over S and so

-

[Der &Ifm(E) : S] = [Der E : S]

+ [Inder -W,,(E) : S].

(1.8.6)

(Der C)s. Hence [Der E : S]= [(Der C ) s : By 1.8.2, Der E = Der Cs S] = [Der C : F].It is clear also that Inder As/S -. (Inder A / F ) s and hence [Inder d/In(E) : S] = [Inder As : S] = [Inder A : F].

(1.8.7)

By (1.8.5). (1.8.6) and (1.8.7) [Der A : F] = [Der C : F]

+ [Inder A : F].

(1.8.8)

We have the exact sequence

0

i

Inder A

+ DerFA

v 11x1 7 -+ 0

+

+

so [DerFA : F] = [Im 7 : F] [Inder A : F]. Since Im 17 c DerFC it follows from thi.; and (1.8.8) that Im r/ = DerFC and hence (1.8.4) is exact.

It is evident that the foregoing theorem implies that every derivation of C I F can be extended to a derivation of A / F .

1.9. Some Tensor Product Constructions \;lie prove first Proposition 1.9.1. If D is n finite dimensional division algebra over a field F and E = F(E1,. . . ;6,) where the tiare indetermznates then DE is a division, algebra. n/loreove,r; DE is central over E if D is central over F .

-

-

Proof. We have DE (DFIE1 ,...,E,J)E and D F I E ~~1 D[[I, . . . ,[,I. Now D[[1] is a domain and ind~ct~ion shows that D [ t l , . . . ,[,.I and hence DFIE tT1 is a domain. We have an imbedding of DF[< in DF(,c1,,,.,E,) and any element of the latter can be written in the form p ( t l , . . . ,[,)-I f where f E D F ; ~ <,.Iand y E F[E1,. . . .[,I. It follows that D E = DF(<,,.... E7) is a domain. Since it is finite dimensional over E it follows that it is a division algebra. The last statement is readily proved using a base for E/F which is also a base for DE over D. Now let E be a cyclic field over F with Gal E / F = (a)and [E : F] = r . Let [ be an indeterminate and form E([). Fie can extend a to an automorphism a

I. Skew Polynomials and Divisiorl Algebras

33

of E(() io that a $ = ( Then E ( $ ) / F ( $ )is cyclic wit11 Gal E($)/F($) = ( a ) . 1e.l; ran forin the ryclic algebra ( E ($), a. 6) which is central sirrlple over F (6).

Proposition 1.9.2. (E($):a;$) is a diriision algebra.

Proof. TVe could prove this using the Wedderburn norm condition. However. we prefer to use a more elementary argument that is applicable also for differential extensions. We form the twisted polynomial ring R = E [ t ;a ] . We have seen that the center of R is F[z]. z = t r . and the localization R of R at the inonoid of non-zero elements - of F[z]is the cyclic division algebra (E. 5.z) over F = F ( z ) where E = FE and 5 / E = a.6 / F = I F . Since [ and z are transcendental over F we have an isomorphism cu of F ( [ ) / F onto F ( z ) / F such that $ --i z. By (1.9.1). E @ F F($) is a field. Hence the canonical homomorphism of E g~ F ( $ ) onto E ( 6 ) is an isomorphism. It follows that me have an extension of a to an isomorphism cu of E([) onto E = F E that is the identity on E and maps $ -, z. The argument used on p. 20 to characterize generalized cyclic extensions shows that a: can he extended to an isomorphism of (E($). a,[) onto R = (E. 6.z ) . Since R is a division ring so is (E([), a. $). Hence this is a division algebra over F([). The following are some interesting special cases of cyclic fields that can be used in the foregoing construction. 1. E = F(C1,. . . ,<,) where the $, are indeterminates. Let a be the autonlorpllism of E/F that permutes the & cyclically and let I = Inv(a). Then E / I is cyclic with Galois group (a). 2. Assume F conta.ins r distinct r-th roots of unity arid let E be a primitive one. Let E = F ( $ ) , $ an indeterminate, and let 0 be the autorrlorpliislri of E/F such t,hat a $ = E [ . Then Inv(a) = F ( t T ) and E / F ( $ " ) is cyclic with Galois group ( 0 ) . 3. Let F be a finite field. E an r-dimensional extension field. Then E/F is cyclic with Galois group ( a ) ,a the Froberlius automorphism a -i a4, q = lFI. We obtain central division algebras by using these cyclic field extensions in ( E ( ( ) , a, 0. As a consequence of the example 2 we can prove the followirlg

Corollary 1.9.3. Let F be a field containing r distinct r - t h roots of 1 and let and r~ he indeterminates. Let D be the algebra over F ( < ,q ) generated by t w o elements x and y subject t o the relations

E

ujhere

E.

is a primitive r - t h root of 1. T h e n D is a central division algebra over

F(C,7). Proof. TVe consider first the field F ( [ ,u) where u r = [. This is cyclic over F ( [ ) with Gal F ( $ , u ) / F ( $ )= ( a ) where a u = EU. Then E = F ( $ , v , u). ur = [, is cyclic over F ( $ ,77) with Gal E / F ( $ , 7 ) = (a) where ou = EU and, by 1.9.2. (E,a, q) is a central division algebra over F ( $ ,7). If v is an element of ( E , a , 77)

1.9. Some Tensor Product Constrllctions

33

such that vu = ( a a ) ~a, t E , and v r = 7 then (E: cr? 7 ) is generated over F ( [ ,11) by the elements u and 1, such that u" = 6;'v = 17, uu = E U V . Now let F ( E ,i l ) { X .Y) be the free associative algebra over F ( [ ,7 ) generated (freely) by X and Y and let B be the ideal in F(E, v ) { X ,Y) generated by the elements Xr - Y7'- v . Y X - E X Y . Put rc = X B. y = Y B. Then, by definit,ion, D = F((,v ) { X ,Y ) / B is the algebra generated by x and y subject to (1.9.3). We have the homomorphism n of D onto ( E ,a, 7)such that x -. u. y --i '1,. It is readily seen that the elements :c2yZ, 0 5 i ,j 5 r - 1, span D over F ( ( , 17). Hence [ D : F ( [ ,v)]5 r 2 . On the othcr hand, [(E.g ,17) : F ( ( , v)] = r 2 . Hence a is ail isolnorphisrrl and D is a division algebra since (E,cr, v ) is a division algebra. < $

+

+

We car1 now prove

Theorem 1.9.5. Let E / F be a cyclic field with Gal E I F = ( a ) and [E : F'] = r and let D be a finite dimensional central division algebra over F such that D 8~E zs a division algebra. Let ( E(0,a, <) be as i n Proposition I . 9.2. T h e n

is a central di,ulsion algebra over F ( [ ) . Proof. The automorphisrn a has a unique extension to an automorphisnl a of D E = D E F E so that cr 1 D = l o . We form R = D E [ t ; a ]whose center is F [ z ] z. = tr and let R be the division ring obtained by localizing R at the ~rlonoidof non-zero elements of F [ z ] .Then R has center F ( z ) and R as algebra over F ( z ) is the generalized cyclic algebra ( D ,5.z ) where D = F ( z ) D E , E I and 6 I D E = a F(z)= Next wc consider the algebra

over F ( z ) where a here is the extension of CT on E ( z ) that is the identity map of F ( z ) .Since both factors are central simple, so is the tensor product. Now we have a homomorphism of ( E ( z ) .a , z ) onto the subalgebra A = ( F ( z ) E 5, . z) of R and a l~omomorphismof D F ( = )onto the subalgebra B = F ( z ) D of R. It is readily seen that if n E A and b E B then ab = ba and R = AB. It follows that me have a homomorphism of the tensor product (1.9.7) onto R. Since (1.9.7) is simple we have an isomorphism and since R is a division algebra so is (1.9.7). Then (1.9.7) is a central division algebra over F ( z ) . Evidently (1.9.7) and (1.9.6) are isomorphic so (1.9.6) is a central division algebra, over F(E). We prove next

Theorem 1.9.8. Let ( E , a , y ) be a cyclic algebra of prime degree r , D a central division algebra over F such that D E = D 8~ E is a division algebra

36

I. Skew Polynomials and Division Algebras

arid let cr be the extension of a to D E ( a T - ' a ) ( ~ ' - ~ a. . ). a for a E D E . T h e n

SO

that a

I

D = l n . Define N ( a ) =

is a divislon acqebm if and ondy if n,o n E D E exists such that N ( a ) = -/. Proof. SVe form the twisted polynomial ring R = D E [ t :6 1 and the generalized By Theorem 1.3.16 this is a division algebra if cyclic algebra R/R(tr - 7). and only if # N ( a ) for a E D E . On the other hand. the argument used in Y the proof of 1.9.5 can be repeated to show that R/R(tT- 7)= ( D E ,cr; (E. a, y ) ' g F D . Hence (1.9.9) is a division algebra if and only if the norm condition holds.

-,

A/)

These results have analogues in the differential case. Let EIF be a finite dirnensiorlal purely inseparable extension of exponent one. S a derivation of EIF.By replacing F by Const 6 we may assume F = Const 6. Let [ be an indeterminate and let 6 denote the extension of 6 to a derivation of E(() such that 66 = 0. It is readily seen that F ( E ) is the subfield of 6-constants in E([).1% cdn form the differential algebra ( E ( < ) .6, <) that is central simple over F ( E ) . Proposition 1.9.10. ( E ( [ )6! , <) is a division algebra. The proof is identical with that of Proposition 1.9.2. We can now state the differential analogues of Theorems 1.9.5 and 1.9.8. SS7r leave the proofs to the reader. Theorem 1.9.11. Let E / F be purely inseparable of exponent one with [ E : F ] < x;6 a deri~uatlonir1 E S ~ L C I L that Const S = F and let ( E ( [ ) ,6 , E ) be as in, 1.9.1.7. Let D be a central division algebra over F such that D @ F E is a division algebra. T h e n (1.9.12) ( E ( ( ) ,6, E ) F ( ( ) D F ( ~ )

Theorem 1.9.13. Let ( E , 6 ,y ) be a differerltial di'i~isionalgebra of prime degree over F and let D be a central division algebra over F such that D s = D BF E is a diuision algebra and let 6 be the extension of 6 to D E so th,at 6 D = 0. T h e n (1.9.14) (E, 6, 2 ) @ F D

i s a division algebra if and ondy if 7: # V p ( a )for any a

E

DE.

37

1.10. Twisted Laurent Series

1.10. Twisted Laurent Series Let R = D [ t ;a ] the t,wisted polynomial ring defined by the division ring D a.nd the automorphisrri a in D . TVe have the ideal I = Rt = t R in R such that nI?'= 0 and so we can define an I-adic Hausdorff topology in R whose open sets are the unions of t,he cosets a In,a E R;n 1 ( B A 11: Sect,ion 7.18). R is a topological ring relative to this topology. The completion R is the ring D [ [ t a; ] ]of formal power series

+

>

where the a, E D . addition is termwise and multiplication is defined by

where

ci, =

ai(aibj).

TVr call R = D [ [ t a: ] ]a tzuzsted power sertes rrng. If a = Ca,tz as in (1.11.1) wc define the order o ( a ) by o(0) = oc,o ( a ) = k if a, = 0 for z < k and ak # 0. Then

The first of these implies that D [ [ t a; ] ] is a domain. If o ( z ) element 1 z z 2 . . . is defined in D [ [ t a: ] ]and

+ + +

> 0 then thc

It follows that the units of D [ [ t a; ] ]are the elements a such that o ( a ) = 0. That this is a necessary condition t,hat a is a unit is clear from the first equation in (1.11.4). To see the converse: let a = a0 a l t . . . with a0 # 0. Then a o l a = 1 z where z = a i l a l t .... Hence a has t,he left inverse (1 - z z2 - . . .)no1.Since we are in a domain this is a two-sided inverse and hence a is a unit. We can localize D [ [ t (TI] ; at ( t ) t,o obtain a division ring. For this purpose we consider the product set D [ [ t a: ] ]x ( t ) where ( t ) = { t k 1; 0 ) . We define ( a ,t k ) (b.t" for a , b E D [ [ t a: ] ]by atE = bt"'. This is an equivalence relat,ion. We denote the class of ( a ,t E )by a/t%nd the quotient set of equivalence classes by D ( ( t :a ) ) . We can make this into a ring by defining

+

+

+

+

+

>

-

a / t V b/te = (at"

btk))lk+'

(a/t"(b/tc) = a(apkb)/tk+'.

(1.10.6)

It is readily checked that the sum and product are well defined and endow D ( ( t :a ) ) with the structure of a ring. The map a --, a l l is a monomorphism of D [ t :a ] into D ( ( t ;a ) ) . We can identify D [ [ ta; ] ]with its image and regard

38

1. Skew Polyriomials and Division Algebras

D ( ( t :0))as an extension of D [ [ ta; ] ] .The elenlent t (= t l l ) has the inverse t p l = l / t in D ( ( t ;a ) ) a i d every elerrlent in D ( ( t ;a ) ) can be written in t,he forin a t p k , a E D [ [ t a: ] ]k: 2 0 , a i d we may assuine o(a) # 0 if atp" 0.Since such a n a is invertible in D [ [ ta; ] ]it is clear that a t f k is a unit in D ( ( t ;a ) ) . Hence this is a divisioii ring. We call D ( ( t :a ) ) a twisted Laur,en?tsrries dzvision ring. Now suppose [ D : C ] = rz2 < x for C the center of D and a I C is of finite order r . Then [ D : F ] = n% for F = C n Inv(n) and nT = I, where we may assume a u = u. The center of R = D [ t :a ] is F [ z ]where z = u-It". Now D [ t :a ] is derise in D [ [ ta; ] ] .Hence F [ z ]c Ccnt D [ [ t a; ] ] .Moreover: the ] formal power series in z with coefficients closure of F [ z ]is t,he ring F [ [ z ]of in F . Hcnce F'[[z]jc cent D [ [ ta; ] ] .It follows also that the ring F ( ( z ) )of formal Laurent series in z with coefficielits in F is contained in tmhecenter of D ( ( t ;a ) ) . MTe also have the division subrirlg D ( ( z ) )of formal Laurelit series in z with coefficients in D . The automorphism a in D can be ext,ended to a n al~tornorphismin D ( ( z ) )so that a z = z. Theorem 1.10.7. Let D be a division ring that is ,finite dimer~sionalover its center C witir, [ D : C ] = n 2 and let a be an automorphism of D such that a C has finite order r . Let u be an element of D such that a r = I, and nu, = u,. F = Cn Inv(a). Then the center of the twisted Laurent series division ring D ( ( t ;a ) ) is F ( ( z ) )where z = u p l t T and D ( ( t ;a ) ) is the generalized cyclic algebra ( D ( ( z ) )a,: u z ) = D ( ( z ) [t: ) o ] / D ( ( z ) ) / [at];(tT - u z ) .

Proof. I+'e have the relations

xz-,,

where d = d,zi, di E D . These imply that we have a homomorphism of the generalized cyclic algebra ( D ( ( z ) )a: ! u z ) / F ( z )into D ( ( t ;a ) ) .This is a, inonomorphism since ( D ( ( z ) )a, , u z ) is simple. Since t-,, can be written in the forrn z-'tm for k , !: m 0 our homomorphism is also surjective. Herlce it is a n isomorphism.

>

1.11. Differential Laurent Series If we attempt to carry over the construction of twisted power series and twisted Laurent series that we considered in the last section to the differential case we run into the difficulty that the ring D [ t ;61 may be simple and hence me cannot introduce a non-trivial I-adic topology in this ring. A way out of this difficulty has been suggested by Cohn [77].In this we replace D [ t ;61 by a related ring. We work 111 the Ore-IVedderburn quotient division ring D ( t . 6) of D [ t ,S ] Let g = t-I E D ( t ;6 ) . Let D [ y ] denote the subring of D ( t ;6) generated by D and y and let I be the ideal in D [ y ]generated by y. We wish to show that D [ y ] / I-. D and nIn = 0 This will permit us to introduce power series in

1.11. Differential Laurent Series

39

y with coefficients in D. Instead of proving these assertions directly we shall follow a different approach based on matrices. TVe note first that the basic commutation formula ta = a t 6a. a E D . gives the formula (1.11.1) ay = ya y(6a)y.

+

+

TVe prefer to write this with 6 replaced by -6 arid we write -6a = a', a" = Then we have (a')'. . . . a('%)=

.

which car1 be iterated t o

If power series were available we would have

T7e now introduce the ring T,(D) of triangular matrices

{EL,,

a,,e,,} and n J n = 0. It is readily seen that This contains the ideal J = T, = T,(D) is complete in the J-adic topology. \Ye define

These are colitaii~edin

T, and for any a, E D, the series

that a - 6 is a monomorphism of the division ring D ( i ) into T,,LIoreover, yii ZLij ijZL1y froin which we can deduce

converges in the J-adic topology. It follows from the Leibniz formula: ( ~ b ) ( = ~)

-

=

This implies that the set of power series (1.11.6) is a subring of T,. It is clear also that CZL,yZ = 0 if and only if every tii = 0. We now simplify the notation and write y for fi: a for ZL arid we denote the ring of power series C a i y q y D [ [ Y611. ~

Theorem 1.11.8. The ring D[[y;611 is a local domain with Jacobson radical K = D[[y;611~.

40

I. Skew Polynomials and Divisiorl Algebras

Proof. Let a @ K. Then a = cr,(l - z ) where (1, # 0 in D and z E K. Since a, is irivertible and 1 - z is invertible with inverse 1 z z 2 . . ., a is a unit. Evidently K 3 1. Ilerice the noil-units constitute the idcal K and D [ [ y 6:11 is local with K as radical. Any elemcnt of D [ [ y 0;11 has the forin ayn where a is a unit and n = 0 . 1 . 2 . . . . &lid for any unit a. ya = Ey where 6 is a ~ l n i t .It follows readily that D [ [ y 6;11 has no zero-divisors # 0.

+ + +

We can localize D [ [ y 6;11 with respect to the moiloid S = ( y ) to obtain a division ring D ( ( y ;6)): the ring of dzfferential Lawent series (orjer D relative t o 6 ) . Thc relation ay = ya y(6a)y gives y-'a = a y p l 60. It follows that we have a inonomorphisnl of D [ t ;61 into D ( ( y ;6 ) ) fixing D arid serldirig t --i y-l. This can be extended to the Ore quotient division ring D ( t ;6 ) and this perrnits us to identify the subring of D ( t :6 ) generated by D arid t-I with the subring D [ y ]of D [ [ y 611 ; generated by D and y. It is clear that the ideal I iri D [ y ]generated by y is contained in I' = K n D [ y ]. It follo~vsreadily that D [ y ] / IT" D and nI'" 0 as we indicated at the beginning.

+

+

11. Brauer Factor Sets and Noether Factor Sets

In this chapter we c,onsider the main classical results of the struct,ure theory of central division algebras and more generally of central simple algebras over arbitrary fields. These center on two closely related problems: the determination of the algebras and the structure of the Brauer group B r ( F ) of a field F. SVe shall begin our discussion with a general collstruction of a central simple algebra A of degree n from a commutative Frobenius subalgebra K with [K : F] = n and anot,her element .c of A. We show that ZJ can be chosen so that A = K c K . Specialization t,o the case in which K is a commutative separable suhalgebra gives rise to Brauer factor sets with values in the multiplicative group E x of a splitting field E of K. Tlie Brauer factor sets define a certain cohomology group H 2 ( K / F ) which is isomorphic to a subgroup of B r ( F ) . If K is a separable ext,cnsion field this subgroup is the subgroup B r ( K / F ) of B r ( F ) of classes of central simple algebras over F split by K . Specialization to the case ill which K = E is Galois over P gives Noetlier factor sets and the fundanlental isomorphism of B r ( E / F ) with t,lie cohomology group H 2 ( G ,E*) where G is the Galois group Gal E/F and the action of G on E' is the natural one. \;lie use crossed products to show t,hat Br(F) is a t,orsion group and to derive the relations between the index and exponent of central simple algebras. I r e give an example due to Brauer to show that these relations are exact. After this we derive results of Wedderburn; Albert and Brauer on central division a,lgehras of degree 5 5. We then return to the general theory t,o derive the results on "inflation" and "restriction" for crossed products. The former leads to an isomorphism of the full Brauer group B r ( F ) with a cohornology group of t'he Galois group G of the separable algebraic closure of F and an isomorphism of the e-torsion part Br,(F) of Br(F) with a cohomology group H:(G, p,) where p, is the set of the eth roots of 1 (assuming char F e)

+

N. Jacobson, Finite-Dimensional Division Algebras over Fields , © Springer-Verlag Berlin Heidelberg 1996, Corrected 2nd printing 2010

42

11. Brauer Factor Sets and Noether Factor Sets

2.1. Frobenius Algebras

Definition. A finite dimensional associative algebra A over F is called a Frobenzus algebra if A contains a hyperplane H that ~ o n t a i n sno noii-zero one sided ideal of A . If i is a linear function on A . i defines a bilinear form !(a. b) = i ( a b ) which is nssociative in the sense that &(ac,b) = l ( a , cb). Any associative bilinear form i ( a , b) on A is obtained in this way since if ! ( a ) = i ( a , 1 ) then i ( a . b) = !(all, 1 ) = L(ab). Let A be Frobenius, T a linear function whose zeros constitute the hyperplane H that contains no one-sided ideal # 0 of A. If r ( a . b) is the bilinear form of r t,hen the set of z such that r ( a >z ) = 0 ( ~ ( az ), = 0 ) for all a t A is a left (right) ideal contained in H. It follows that A is Frobenius if and only if there exists a n associative non-degenerate bilinear form on A. 1 e note also that if A is Frobenius and ~ ( ab), is as indicated then for a left ideal I, the subspace I ' ~= { c / r ( I , c ) = 0 ) is the right annihilator a n n R I of I. Hence this : F] = n - [I: F].Similarly, if I is a right ideal then is a right ideal and [ I ' ~ 111,= {c / ~ ( cI): = 0) is the left annihilator of I and [ I ' ~: F] = n - [I: F]. It follows that the map I --, I ' ~ is a lattice anti-isomorphism of the latt,ice of left ideals of A ont,o the lattice of right ideals. If .4 is Frobenius then AE is Frobenius for any extension field EIF. If A = Al 3 . . . @ A, where the Ai are ideals then A is Frobenius if and only if every A, is Frobenius. We also have

Proposition 2.1.2. If A and B are Frobenius algebras then A 8~ B is a Frobentus algebra. Proof. Let T and a be linear functions on A and B respectively such t,hat the corresponding hyperplanes contain no non-zero one-sided ideals. Let cp = T @a be the linear function on A@B such that y ( a @b) = r ( c ~ ) a ( bfor ) a E A, b E B. Let z = C a i g bi; ai E A, b, E B , satisfy p ( z c ) = 0 for all c E A 8 B. We niay assume the a, are linearly independent. Then there exist x, E A such that r ( a i x j ) = S i j . Let y E B and put c = x, @ y. Then the condition cp(zc) = 0 gives a ( b i , y) = 0 . Since this holds for all y , bi = 0 for all i and hence z = 0. Similarly ~ ( c z = ) 0 for all c implies z = 0 . It follows that the bilinear form p ( c , d ) = ~ ( c don ) A @ B is non-degenerate. Hence A 8 B is Frobenius. 14%shall derive next a condition that a commutative algebra A be a Frobenius algebra. \VE can write A = A1 @ . . . $ A, where the A, are ideals and are local algebras (BA 11. p. 111).For such an algebra we have

Proposition 2.1.3. A local commutative algebra is Frobenius if and only i f i t contains a unique minimal ideal.

2.2. Cornmutative Frobcni~isSubalgebras

43

Proof. Let A be a local colnnlutative Frobenius algebra. Since A contains a unique lllaxiinal ideal it follours from the anti-automorphism of the lattice of ideals of A that A contains a unique mininlal ideal. Conversely, suppose A is a conimutative algebra that contains a unique mini~nalideal N. We can choose a hyperplane H not containing AT. Tllen H contains no non-zero ideal of A a.iid hence A is Frobenius. Corollary 2.1.4. Any algebra A = F [ a ] ~rlithn sir~glegenerator Is Frobe?ai?ss.

ns

Proof. Let f (A) he t,he ininirnum polynomial of a and let f (A) = P,(A)~& where the p,(A) are distinct primes in FIX]. Then A = F [all 9 . . . 9F [ a s ] wherc the minimum polynolnial of ai is p,(A)".. Now every ideal # 0 in F [ a , ] contains the ideal generated by b, = pi Hence F[a,] contains a unique rninimal idea.1 and so this is a Frobenius algebra. Then F [ a ] is Frobenius. The following module result will play a key role in obtaining a special type of generation of a central simple algebra by a commutative Froberiius subalgebra.

Proposition 2.1.5. Let A be a commutative Frobenius algebra and let M be a faithful A-module. Then, ill con,tain,s a s~l,bmoduleisomorphic t o A. Proof. Write A = Al 0 . . . 3 A , where the Ai are local (BA 11, pp. 425427). Then Ai contains a unique mirliina,l ideal N ? . We have M = All1 = AlAl 8 . 8 A s M and A i M is a faithful Ai-module. Hence there exists a n x, E A,M such that A$zi # 0.The map ai --i aixi is an Ai-homomorphisn~of A, onto A,x, which is a submodule of A,M. The kernel of this honlomorphisnl does not contain Ni. Since Ni is contained in every non-zero ideal of Ai the kernel is 0. Hence A,x, -. Ai and Alxl @ . . . $ Asxs is a submodule of M isomorphic t o A.

2.2. Commutative Frobenius Subalgebras Let A be central simple. We can regard A as Ae = A @ A' module where AO is the opposite algebra of A by defining

where a , . x E A, b, E AO (= A as vector space). Now A' is simple since A and AO are central simple. Hence A is a faithful Ae-module (that is, the corresponding representation is faithful). If IT is a subalgebra of A then restricting tllc action of Ae t o K e = K g KO rnakes A a K'-module and. of course. this is faithful. We now let K be a commutative Frobenius subalgebra whose dimensionality is the degree of A. Then we have

44

11. Brauer Factor Sets and Nocther Factor Sets

Theorem 2.2.2. Let A be a central simple algebra of degree n ; K a commutative Frobenius subalgebra of A such that [ K : F ] = n . T h e n A as K e = K @ I < module (KO -. K ) is isomorphic t o K e an,d hence there exists a v E A such that A = K c K .

Pmo,f. The algebra K " is a corninu~at~ive Frobenius algebra and A is a faithful K'-module. Hence. by 2.1.5, A corlt,airis a submodule isomorphic to K e . On t,he other hand: [A : F] = "n [K" : F ] . Hence A -. K e as Ke-module. Since K e is cyclic with 1 X 1 as generator: 4 is cyclic as K"-module. If c is a generator of this module then A = K e v = KvK. We shall require also Theorem 2.2.3 (Jacobson [ 7 5 ] ) . Let A and K be as i n 2.2.2. Then: (1) I< = A K , the centralizer of K i n A, (2) any isomorphism of K into A can be extended t o a n inner automorph,ism of A, (3) any derivation o,f K into A can be extended t o a n inner derivation of A.

Proof. ( 1 ) Let L = EndFK. Then we have the isomorphism k --t kL (n: --t kn:) c ~ fK into L and we can identify K with it,s image K L ill L. Since K is colliniutative and the centralizer of the set of left multiplications of an algebra is the set of right multiplications, LK = K . On the other hand. can be characterized by the module condition A" = { c € A / (1 8 k - k % 1 ) c = 0. k E K} and since A and L are Ke-isomorphic by 2.2.2 and L~ = K ; we have A" = K . (2) We suppose first that A = EridFV where V is an n dimensional vector space over F. Let a be an isomorphism of K into A. We may regard V as K-module in two ways. In the first the action is the natural action of K as subalgebra of A and in the second it, is the composite of the isomorphism g with the natural action. Since V is faithful as K-module under both actions and [V : F] = ,n,it follows from 2.1.5 that the two K-modules we have defined on V are isomorphic. This means t,hat there exists a bijective linear t,ransformation u of V such that for every k t K , a ( k ) = uku-l. Then u E A and a call be extended to the inner autornorphism I, in A = EndFV. Kext let A be arbitrary. If A is split the result is covered by the case A = EndFV. Hence we may assume A is not split and so by Wedderburn's theorem on the coinmutat,ivity of finite division rings the base field F is infinite. If F is the algebraic closure of F then AF is split and K F is a commutative Frobenius subalgebra of A F . Applying t,he result in this case we obtain an invertible elenlent 21 € A F such that o(k)21 = iik! k € I<. NOWlet (e,. . . . . e,,n) be a base for A / F and hence for A F / F and let ( k l , . . . ; k,) be a base for K I F . Then we liave ti = w m , we E F , and the conditions that i~(k,)21= ilki are equivalent to a set of homogeneous linear equations on the we with coefficients in F . It follows that if U denotes the F-space of A of elements u such that cr(ki)u = I L ~ , .1 < i n, and 0 the F subspace of AF of elements ii such U p . The subset of invertible elements of that a(k,)21 = iik,, then u = FU

~7

r

<

-

2.3. Brauer Factor Sets

45

U is the open subset defined by n(!u) # 0. Since the corresponding subset of u is not xracuous it follows that we have invertible u in A such that n ( k , ) u = u k , and hence such that o ( k ) u = u k or a ( k ) = ' u k u p l ,k E K . Thus 0 can be extended to the inner automorphism I, of A. (3). Let 6 be a derivation of I( into A. Then 6 E L = EndFK and the condition 6 ( k l ) = k ( @ ) (6k)L for k . L E K is equivalent to the operator . suppose S --i d and 1 -, v under a kecondition S k L = kL6 l ( 6 k ) ~Now isomorphism of L into A. Then we have d k L = k L d u ( ~ Kand ) ~v E k since l k L = KL1 implies that v E AK = K . illoreover, l K L = K L gives c K which implies that 1: is invertible. Hence we haye ( d v - ' ) k L = k L ( d v p l ) ( 6 k ) so ~ 6 can be extended to the inner derivation x -+ (dv-l)n: - z ( d v - l ) in A.

+

+

+

+

2.3. Brauer Factor Sets Let A be central simple of degree n. \Ve have shown in 1.6.20 that if the base field F is infinite then A contains an element u such that the minimum polynomial f ( A ) is of degree n with distinct roots. It is readily seen that the same result holds for finite A. For, in this case. .4 -. h f , ( F ) and we can choose f ( A ) to be an irreducible polynomial of degree n over F. Then h f n ( F ) contains a matrix ~vlloseminimum polynomial is f ( A ) . Now let K = F[u] be a subalgebra of A such that the minimum polynomial f ( A ) over F of u is of degree n wit>h distinct roots. Then K is a commutative Frobenius subalgebra of A and hence, by 2.2.2. A contailis an element v such that A = K v K . Let E be a splitting field over F for f ( A ) so E = F ( r l , r z , . . . , r,,) where the r , are distinct and f ( A ) = 17(A - r , ) in E [ A ] . Consider the algebra K E = E [ u ]= E [ A ] / ( f( A ) ) . In this algebra we have n non-zero ortllogorlal iderrlpoteilts

e, =

( u - r l ) . . . ( u - r,-1)(u ( r , - r ~ j . , . ( r ,- r t - l ) ( r 2

r l + l ) . . . ( u - r,) - r l + 1 ) . . . ( r ,- r n )

-

(2.3.1)

such that C e , = 1 (see e.g. BA 11, p. 479). Hence K E = @: Ee,. Now AE is central simple of degree n over E and K E co~ltaiiisthe n orthogorlal idempoterlt,s ei. It follows that AE = l1)fn(E)SO E is a splitting field for A. Thus we may regard A as an F-subalgebra of A f 7 , ( E ) such that E A = Af,,(E). Also we inay suppose that (2.3.2) u = diag(r1, r z , . . . ; r,). then ukz1uE = ( r f r $ v i 3 ) and since A = K ' v K the elements 0 5 kk.. t 5 n - 1: form a base for A I F . Hence every element of A is a matrix L = (&Jvz3) (2.3.3) If r =

[zliJ)

u%,u<

where

71

40:

11. Brauer Factor Sets and Noether Factor Sets

and the ake E F and are uniquely determined. Since E A = A& (E) it is clear that every v,, # 0. Let G = Gal E/F.If a E G:o r , = r,/ and a is determined by the denote this permutation by a also, so per~nutationi i' of {I: 2 . . . . n } . n: is defined by (2.3.4) then the Pig satisfy me have ar, = r,,. If L i J , 1 5 i: j

-

.

<

These conditions, which we shall call the conjugacy condatzons on are also sufficient that the E,, have the form (2.3.4):for we have

t

=

(l,,).

Lemma 2.3.6. Let l = ( l t j )he n matrix of elerrrents tTJE E satisfyin,g the conjugacy conditions (2.3.5). Then there exist ak. E F such that (2.3.4) holds ,for all i,j. Proof. Let V be the Vandermonde matrix

Then I/ is invertible. Hence there exists a unique matrix a = (a,,) E A I n ( E ) such that V U ( ~ V=)P. (2.3.8) This rrlatrix relation is equivalent to the equations (2.3.4).Applying a E G to these equations we obtain

or t,, = C k , e ( ~ ~ l c u ) r ~ - lBy r ~the - l uniqueness of a we have oak! = aka for every a E G. Hence ake E F. (The foregoing proof is due to Walter Feit.) We now put L = (P,,vij), L' = ( k $ J ~ i jwhere ) the tZJ and tij satisfy the conjugacy conditions, so L, L' E A.Then LL' = L" = (!:\v,~) where

Lemma 2.3.11. The c,jk satisfy

2.3. Brauer Factor Sets

47

Proof Apply rr to (2.3.9) to obtain C:, = Po, ,k(ac,k,)e~k,uJ.On the o t h e ~hand. l : = P,,,,kc,L,o~,~,e',~ Hence we have

,,

,,,,

Lik

and these relations hold for all E t k . hatisfying the conjugacy conditions. TVe can also write (2.3.15) C k r ~ , k j(uk, . I;,) = O. e,k, = &I 7 ' 2 . k

Now A J n ( F ) = EA. Hence taking a suitable E-linear combination of the matrices irl A we obt,ain a mat,rix whose j-t,h column is ( 0 . . . . , 0 , l :0 , . . . , 0 ) where the 1 is in any chosen position. Usirlg this linear colnbirlation of the relations (2.3.15) we obtain l i k e z k , = 0 for all k . Then ezk, = 0 and dikj = 0 which is (2.3.12). Now (2.3.13) follows by direct verification using the definition (2.3.10). \Vc now define a Brauer factor set c to be an indexed set of elements cZjk t E" 1 L . 1. k 5 n. such that

<

C?jkCikB

= Ci31c3kY.

(ii)

The foregoing lemma states that the c,,k defined by (2.3.10) from the element z1 = ( u , , ) constitute a Brauer factor set. We shall call (i) the conjugacy conditions on the c,,k. We note that these imply that

ctgic E F ( r , , r 3 ,r k )

(iii)

For, if a E Gal E / F ( r , , r j , r k ) t,hen ai = i , a j = j , a k = h- and hencc, by (i), ocyk = e,jk Since this holds for every a E Gal E / F ( r i , r , , r k ) , c,,k E F ( r i ,r,, r . k ) b y the Galois correspondence. If we put i = j = k and j = k = P successively i r ~(ii) we obtain

JYe notc also that if we take c,,k = 1 for all 2 . J . k then we obtain a Brauer factor set. This Brauer factor set will play a distinguished role in the sequel and will be denoted as 1. We have seen that if we define c,,k = ~ , , v , ~ vthen , ~ ~c = {c,,k) is a Brauer factor set. Here v = (v,,) is any element of A such that A = K v K . We now provided that observe that c is independent of the imbedding of A in Mn(E) u = diag{rl, 1-2, . . . . r,) in the imbcdding. For, if we have a second imbedding with this property then it follows from the Skolem-Noether Theorem and the

48

11. Brauer Factor Sets and Noether Factor Sets

fact that the only matrices that commute with u are diagonal matrices that in the second imbedding we have v = (d,v,,d,l) mrhere d , E E M .Then

JITe shall now normalize r_. so that, the corresponding factor set c is reduced in the sense that every ci,, = 1. B y (iv) this irrlplies c,,, = 1 = c,,, for all i . j . JVe remark that if f ( A ) is irreducible or: equivalently. K is a field then c is reduced if = 1. For. in this case the permutation group of t,he T , deterniined by G is transitive. Then ell1 = 1 implies c,i, = 1. JVe now note that? by (2.3.10), c,ii = v,, so a7:,, = G,,, a E G. Hence if me put tii = o i 2 , and OT3 = 0 if i f j : then the conjugacy coiiditions hold for the f Z j so Since y commutes with u, y E F [ u ] and we can replace v by yv. This normalization permits us to assume ?', = 1 and hence c,,, = 1 , that is, c is reduced. TVe can now prove

Theorem 2.3.17. Let K = F[u]be finite dimensional separable, , f ( A ) the m i n i m u m polynomzal of u over F and let E = F ( r l : . . . ,r,) be the splittang ,field o f f ( A ) over F wlzere f ( A ) = 17(A - r i ) i n E [ A ] . Suppose c = { c i J k ) is a reduced Brauer factor set with vo,l?~esirr E x a,nd let B ( K ,c ) denote the subset qf 1221n(E)o,f mat rice.^ != ( t Z Jsu,cl~ ) th,at a&, = l,,.,,: cs E G = Gal E I F . T h e n B ( K .c ) is a n F-subspace of M n ( E ) ,and if we define a c-product !,tl for O = (O,,), i' = (Pi,) E B ( K ,c ) as E'I = (ti;)'where

t h e n B(K, c ) becomes a central simple associative algebra of degree n over F con,tainin,g a subalgebra isomorphic t o K . &loreover, the m a p

e = (&,)

-a+

L=(

C ~ ~ I ~ , ~ )

(2.3.19)

i s a n isomorphism of B ( K ,c ) 'with a n F-subalgebra A of &fn(E).Conz~ersely, every central simple algebra of degree n containing K as subalgebra can be obtained by this construction. Proof. It is clear that B = B ( K ,c ) is an F-subspace of A I n ( E ) and if defined by (2.3.18) then

!:/,is

2 . 3 . Brauer Factor Sets

49

Hence B is closed under the c-multiplication. Consider the map defined by (2.3.19).Evidently this is F-linear and injective. The (i, 9)-entry of the matrix product ( ~ 7 , 1!J, ) (cz3dl,,) is

Hence the map is a homornorphisrn for multiplication. The irnage A = {L) of B is an F-subspace of Afr,(E) closed under multiplication. Observe next that since ci,, = 1, any diagonal matrix satisfying the conjugacy conditions is fixed under (2.3.19). Then 1 E A and 1 is the unit of A and of B. Hence A is an P-subalgebra of hf,, ( E ) . We note next that r = diag{rl, 7-2, . . . ,r,) E A and F [ r ] is a subalgebra of A isomorphic to K . Next let !,j = 1 for all i , j and let s be the corresponding matrix (cijlCi,) = (cljl). Note that every entry of s is # 0. Now every matrix unit e,, E E [ r ] ,and since ei,sej, is a non-zero multiple of eij it is clear that, E A = M n ( E ) . Since l\Jn ( E ) is simple A contains no nilpotent ideals # 0 and A is not a direct sum of more than one non-zero ideal. Hence by the Wedderburn structure theory, A is simple. Any element of the center of A is in the center of M n ( E ) and so is a scalar matrix. Such an element has pre-image under (2.3.19) that is a diagonal matrix diag{el,. . . .!,,). The conditions , ! a = emi arld diag{il, . . . En,) is a scalar matrix imply that this element is in F1. Hence A is central simple. Then & ( E ) -. E E F A (BA 11, Theorem 4.7; p. 218) and consequently A has degree n . The isomorphisnl of B ( K , c) with A 1zourimplies that B(Iz', c) is central simple of degree 12 and B ( K , c) contains a subalgebra isomorphic to K . Conversely, assurne A is central simple of degree n containing K = F [ u ] . We have seen that A = K r K and we can identify A with the F-subalgebra of matrices ( ~ , ~ e , ,where ) I) = (wa3) has all its entries # 0 and (L,,) satisfies the 1 conjugacy conditions. hloreover, if we define c,k, = cikvk.zl,, then c = {c,,~) is a Brauer factor set,. By normalizing v we may assume c is reduced. Now we have (v,,L,,)(~~,~!:~) = (v,jl;;) where Ei; is giver1 by (2.3.18). Hence the map (!,,) (rL.23Ez3) is an isomorphism of (B,c) onto A.

-

V'e shall now determine the elements u1E A such that A = K w K . We cldirn that these are just the elements w = ( E 2 , ~ , L 3 ) such that every !,J # 0. SVe have seen that E [ u ]= D = CEe,,. It is clear that D7uD = M n ( E ) for a rrlatrix 711 = ( u J ~ if, ) and only if every ul,, # 0 On the other hand, D = E K and hence DmD = E K w E K = E K w K . No~vif 7 0 E A then K u ' K c A and hence K w K = E ZFK w K . Hence A = K w K for u. E A if and only if u l = (!,,vLJ) with every # 0. \Ye have associated with an element v E A such that A = KwK a factor sct c = {cLJk)where ~ , , k = ~ , ~ v , for i ' I) = (t),,). If u l = ( t , , ~ ) ,where ~ ) the eZJ satisfy the conjugacy conditions and every l,]# 0 then the Brauer factor set determined by 71: is c' = {clJk) where

!,,

50

11. Bra~ierFactor Sets and Nocthcr Factor Sets

c:7k = ! t J ~ 3 k ~ z < 1 ~ ~ l l k .

(2.3.20)

Two Brauer factor sets related in this way by L,, satisfying the conjugacy conditions are called assoczates. These constitute an equivalence class. \n;p denote the equivalence class of the Brauer factor sets all of whose ct3k = 1. by 1 and the relation of associateness by --.

Theorem 2.3.21 Th!e algebras B ( K , c) an,d B ( K , c') are isomorphic under a inap which is the identiby on K = F[diay{srl,. . . , r,)] if and only if c and c' are n,ssociated (reduced) factor sets. for all i; j; k , where ami, = 'm,,, , Proof. (Seligman) If c:,~ = ~,,krn,,rn~krn,;~ then (!,j) --t (.eijm,jl) is an isomorphism of B ( K , c) onto B ( K , c'). Because c arid c' are reduced, all m,i = 1, so it is the identity on K . Conversely. suppose B ( K , c) and B ( K , c') are isornorphic by a map extending the identity on K . Note that in the proof of Theorem 2.3.17, the isomorphism (2.3.19) could be replaced by (F, : (!ij) --t L = (~i,,!,~), for each fixed index s: 1 5 s 5 n,. We have a corresponding map of B ( K : c') into A.ri,(E), also denoted p,. Thus: for each s , p,(B(K,c)) and (pS(B(K,cf))are isomorphic F-subalgebras A, A' of Afn (E),with E @ A= 1bf7,(E) = E @ A f .The isomorphism of A and A' resulting from the original isomorphism of B ( K , c) and B ( K , c') and the maps (F, thus extends to a unique E-linear isomorphism Dl,( E ) + A& ( E ) fixing E % K , the diagonal matrices in A& (E). Therefore there is an invertible diagonal E-matrix d ( " ) = d i a y { ~ ? ) ,. . . for each s. such that

A?))

for all (!,) E B ( K , c), where (tiJ) is the image of (t,,) under the given isomorpllism B ( K , c) + B ( K , c'). In particular, when ti, = 1 for all 1 , j , the pre-irnage (rnzJ)E B ( K . c) satisfies (s)-l

c2.7' S = Xi

nvi,c,,,h~") for all i,j, s .

Setting s = j gives 1 = ~:')~'ln,,i:'), or rnI3 = hl3)hj")-l.Fronl the above,

so c and c' are associated.

2.1. Condition for Split Algebra. The Tensor Product Theorem.

51

2.4. Condition for Split Algebra. The Tensor Product

Theorem. lye retain the notations of the last section. We prove first

--

Theorem 2.4.1. B ( K , c) 1 in, the Brauser group B r ( F ) (that is, B ( K , c) Al,(F))if and or~lyl,f c 1.

-.

P'roof. Suppose c -- 1. Then we may assume every c,,k = 1. Hence the subalgebra A of A&(E) isomorphic to B ( K , c) contains the matrix all of whose ent,ries are 1. This matrix has rank 1 and hence the left ideal *ildn(E)?? of hrl,(E) is srlinimal a.nd so is n dimensional over E. It follows that [Av : F] = n. Then A has a representation by n x n matrices over F determined by the module Av. It follo~vsthat A -. f\.fl,(F). Conversely, suppose B ( K , c) 2: Af,(F). Then B ( K : c) B ( K , 1). We have shown in 2.2.3 t,hat if A is central simple of degree n and K1 arid hT2are isomorphic commut,ative Frobenius subalge: F] = n then any isonlorphism of K1 onto K 2 can be bras of A with extended to a.n inner automorpllism of A. Hence if B ( K , c) -. B ( K , 1) then we may assume that we have an isomorphism between these algebras that, is the identity map on K . Let A1 and A2 be the subalgebras of fL.I,,(E)isomorphic to B ( K , 1) and B ( K , c ) respectively by (2.3.19). Then Ai contains the matrix u = diag{rl, r z : . . . , r,), A1 contains the matrix 111 all of whose ermies arc 1 and A2 contains v2 = ( u t j ) SO that ci3k = ~,~l;'~~/-',j~. Then Al = F [ ~ l ] v ~ F [ uAz ] : = F[u]v2F[u]and we have an isomorpl~ism7 of A2 onto Al that is t,he identity on u. Then wl = v(va) satisfies A1 = F[u]wlF[u] and we have seeri that the Brauer factor set determined by wl is c. Since that determined by vl is 1 we have c 1. ' ~ 1

-

-

We consider next the tensor product of two central sirrlple algebras A,. z = 1.2, of degree n containing K = F[u] as subalgebra. Let v, be all element of A, such that A, = Kv,li' and let .I:,= (vi;c))in an imbedding of A, in ildn(E). The algebra A1 8~A:! contains K @ F A2 which contains % F K . We have the sur,jective algebra hosnomorphism

such that Cn, % b, --i Ca,b,. Since K @ F K is semi-simple (which is easily seeri by extending F to its algebraic closure) we have

where e is an idempotent and ( K X F K ) ( 1 - e) = ker u. Then ( K 8 K ) e -. ( K K)/ker u E K . LIoreover, since a 8 1- 1% a E ker v for a E K , we have

We now consider the algebra

52

11. Brauer Factor Sets and Noether Factor Sets

Since A1 and A2 are central simple so is Al E F A 2 and hence so is A. Sloreover, A and A1 B F A2 determine the same element of the Braucr group B r ( F ) and A contains e ( K gF K ) e -. K which we can identify with K . Then we h a w

Theorem 2.4.6. A is of degree n containing K : and a Brauer factor set where } c(') = {c:;)~} is a associated with A as in (2.4.51 is C ( ' ) C ( ~ ) = { c : ~ ~ c ~ : ~ B ~ , a u efactor r set associated with A,. 2.7 a(" = Proof, If a(') = (a!1'),

((a!))

E

AI,(E) we define

and we use this tensor m~lltiplicationof matrices to obtain a n imbedding of Al g F A2 in n/fnz( E ) . Since u = diag{rl, rz. . . . . r,) in Mn(E)it is clear that the matrix for any a E K @ F K in ,Wnz(E) is a diagonal matrix. Hence the matrix for e is diagonal with ~rltriesO and 1. Also we have

Hence the condition (2.4.4) for a = u implies that all the diagonal entries of e are 0 with the exception of those in the positions 1,n 2 , 2 n 3, . . . : n2. This ( E ) e has degree 71 and llerlce t,lle degree of = e ( A l @A2)e implies that is 5 n . O n the other hand. t,his degree is n since A > K. Hence A has degree n and the diagonal entries of e in the positions 1 , n 2 : . . . n2 are 1 and the remaining ones are 0. The matrix e(z.18 z~2)ein n'fnna (E) has non-zero entries only in the ( ( k - 1)n k, ( l l ) n f !): positions 1 k, .t n , and the (2) entry in this position is u,(1) ! vk, . By performing a similarity transformation by a permutation matrix and cut,ting down to a diagonal block we obtain a n imbedding of A in h f n ( E ) in which w = diag{rl,. . . , r,,} and 1. = e(vl 8 t12)e = (ci:)vg)). Since all the entries of v are # 0: n/In(E) = (CEe,,)u(CEei,) and hence A = K,LIK.It follows that we can use v to determine a Bratier factor set for A. Evidently this set is c ( ' ) c ( ~ ) .

<

+

+

+

>

.

+

<

<

2.5. The Brauer Group Br(K/F)

53

2.5. The Brauer Group Br(K/F) From now on we assurrle for simplicity that the base field F is infinite. 4 s before, let K be a finite dimensional cornrnutative separable algebra over F . If F is the algebraic closure of F then K F = F e l @ . . . 8 Fen where the e, are orthogonal idempotents and n = [ K : F ] . It follows that t,he degree of K = deg liF = n. Hence, by Theorem 1.6.21, K = F[u].Moreover, the minimum polyrlorrlial f ( A ) of u is of degree n and has distinct roots. \lie shall say that an extension field E I F splits K if K B = E e l $ . . . @ Ee, where the ei are orthogonal idempotents and we call E a splitting field for I i if E splits K and 110 proper subfield of E splits K. It is readily seen that E is a splitting field for K I F if and only if E is a splitting field in the usual sense for the polynomial f (A). Hence any two splitting fields E I F and E ' I F of K I F are isolrlorphic. Now let E I F be a splitting field for K I F where K = F[u]and ,f ( A ) is the minimum polynomial of 1s. Then E is a split,ting field of f (A).For each root 'r. of f (A) we have a homomorphism a of K I P into E I F such that u --t r. In this way we obtain n = [K : F] homomorphisms of K I F into E I F such that a,lL = r, where f ( A ) = 17(X - ri) in E[A].Moreover, this gives all the liomomorphisms of K I F into E I F . Thus

is the set of homomorphisms of h'lF into E I F . If o E G = Gal E I F then OCY, E A I . In fact, wc have aa,u = ar, = r,, so aa, = a,,. Now let c = { c , ~be~ a) Brauer factor set. Tire can regard this as a rnap c of AI x n/I x h.1 into E* such that

c : (a,:a j :ak) Accordingly, wc write c(ai,a,, a k ) for t,he ciji, are first that

-

ctlk.

ci,jk.

(2.5.2)

Then the defining conditions on

or, independently of t,he indexing,

Vve' shall now call these conditions homogenezty and, more generally. if g : r

;l'f x

. . . x ncr' + E

or E" then g is homogeneous if

for a . 0 . . . . . E E M. In addition to this condition on c we have

11. Bralier Factor Scts and Nocther Factor Sets

54

for a . p. -,. 6 E A1 and c is reduced if c ( a ,a :a ) = 1 for all a E M . This implies that c ( p , w , cu) = 1 = c ( a , a , p ) for all a . 0.If K is a field t,llerl c is reduced if c ( a :a , a ) = 1 for a single a E AT. Similarly, a matrix L = (E,:,) E M,(E) can be regarded as a map (a,,a 3 ) l,,. The usual rnatrix product of l and 1' can then be defined by &Lf(a, P) = C,Eni L(n,-,)E1(?.,O). Honlogeneity of I as map of hl x h1 + E is equivalent = atzj. to the conjugacy conditions We can now re-stat,e Theorem 2.3.17 in the following way:

-

Theorem 2.5.6. Let K / F be a finite dimensional separable commutative algebra, E I F a splitting field for K I F , c a reduced Brauer factor set with values i n E x . Let B ( K , c ) denote the F-space of homogeneozu 'maps of l\f x M into E and define a product i n B ( K , c ) by

for 8 ,!' E B ( K , c ) . T h e n B ( K , c ) becomes a central simple algebra of degree n = [ K : F ] con,taining a subalgebra isomorphic to K . Moreover, for any fixed E AJ the map E --i L where

is a n isomorphism of B ( K , c ) with a n F-subalgebra A of the matrix algebra of maps of 21 x M into E . Conversely, any central simple algebra of degree n con,tainin,g K as a subalgebra can be obtained i n this way. SSTe shall call B ( K , c ) the Brauer algebra determzned by the Brauer factor set c. T h r condition that K is a commutative separable subalgebra of dinlension equal to the degree is equivalent to two other conditions given in

Theorem 2.5.9. Let A be central simple of degree n over F , K I F a cornmutative separable subalgebra of A. Tlzen the following conditions o n K are eqiuivalerrt: (l;) [ K : F ] = n , (ii) K is a maximal cornn~utatiueseparable subalg e h m of A, (iii) th,e cen,tmlizer AK = K .

Proof. (i)+(ii). Suppose L is a commutative separable subalgebra of A containing K . Then L = F [ v ] and the degree of the minimum polynomial of e. deg A = [ K : F] Hence [ L : F ] 5 [ K : F]so L = K . (ii)+(iii). Let K be a maximal commutative subalgebra of A. Then K c A X . Now AK is separable. For, if F is the algebraic closure of F then AF = I ~ I , ( F ) . F = ~ P e l CE . . . @ Fern where the e, are non-zero orthogoKF - ~ [ ~ ( p ) z F e % , na1 idempotents such that C e , = 1. Then ( A K ) FY AF It is clcar that the last algebra is a direct sun1 of algebras !Vn7( F ) . Hence AK is separable. Then the center of AK is separable and since it contains K it coincides with K by the maximality of K. Now AK = A1 @ . . . 6 A, where A, is separable with separable center K , and K1 . . K,. Suppose

<

+ +

2.5. The Brauer Group Br(K/F)

55

for some Ai 3 K,. If A, is not a division algebra then A, contains 7n 2 2 non-zero orthogonal idempotents f j such that C f , = 1, the unit of A,. Then . . . K, is a commutative separable subKl ... C K , f, algebra of A properly containing K contrary t,o t,he ma,ximality of K . The same conclusion holds if Ai is a division algebra since in this case A, contains a separable subfield properly containing Ki. These contradictions show that Ai= K Tfor every i and hence AK = K . (iii)+(i) Suppose AK = K . Then A ": = K F for F the algebraic closure of F and hence n i ~ ~ ( F )=~c iF ~~ ~,where t the e, are non-zero ort,hogonal idempotents such that Cei = 1 and 7 n = [K : F]. It, follows that nl = n and [ K : F] = n.

+ +

+

+

+ +

The Brauer factor sets (regarded as maps of M x M x M into E*) form a group under nlultiplication of images in E * . This contains the subgroup of factor sets such that

where ! : M x M -+ E* is homogeneous. We can form the factor group which we shall denote as H 2 ( K / F ) . If c is a factor set then ! defined by !(a, a ) = c ( a , a. a)-', !(a, 8) = 1 if cu # R is llornogeneous and c(o. p. r ) l ( a ,r'J')!(D, y)P(a, r)-l is reduced. It follows that H 2 ( K .F ) is the factor group of the group of reduced Brauer factor sets with respect to its subgroup of reduced Brauer factor sets of the form (2.5.10). T;lTerecall that an extension field K of the base field F of a central simple algebra A is called a splitting field for A if AK = K @ F A = lVfn(K). If [A] denotes the similarity class of A in the Brauer group B r ( F ) then K is a splitting field for A if and only if it is a splitting field for every B E [A]. Hence me may regard K as a splitting field of the class [A]. We have the homomorphism [A] --i [AK]of Br(F) into Br(K) whose kernel is the subgroup Br(lr'/F) of classes [A] split by K , that is, having K as splitting field. A classical theorem of Brauer and Noether gives a determination of the finite dimensional K that split a class [A]:Let A be a central division algebra over F. Then K splits A if and only if [K : F] = rd where d is the degree of A and K is isomorphic to a subfield of iVIr(A) (Theorem 4.12. p. 224 of BA 11). Now let K be finite dimensional separable over F. Then we have

Theorem 2.5.11. B r ( K / F ) zs a subg~oupof Br(F) isomorphic to H2( K / F ) . Proof. We have the surjective map c --, [ B ( K ,c)] of the group of reduced Brauer factor sets with values in E* onto B r ( K / F ) . Now let c(l) and c(') be reduced Brauer factor sets. Then it follows from Theorem 2.4.6 that B(K. d l ) ) $?F B(K, d 2 ) ) -- B(K. c(')c(')). This implies that B r ( K / F ) is a subgroup of B r ( F ) and c -i [ B ( K ,c)] is a homomorphis~nof the group of reduced Brauer factor sets onto B r ( K / F ) . By Theorem 2.4.1, the kernel of this

56

11. Bralier Factor Sets and Noether Factor Sets

homomorphisln is the group of reduced c

-

1. Hence B r ( K / F ) E H 2 ( K / F ) .

15-e also have the following generalization of the theorein of SpeiserNoethcr that H1(G.E x )= 1 for G the Galois group of E / F .

Theorem 2.5.12. Let K be n finztc dimcnszonal commutatz~iesepnrable alqebra over F. E I F a splzttzr~yfield for KIF, ,I1= { a ) the set of homomorphzsms of KIF ~ n t oE I F . Let ( a , P )--i b(cu, R ) be a hornogenrol~smap of 111x 111 znto E* S U C ~ Zthat (2.5.13) b(a,B ) b ( O , 3) = b(a.? )

for a . 6.7 E J f . Then there esrsts an znvertzble a E K such that

P,roof. Consider t,he Brauer algebra B(I<,1 ) which is the F-space of homogeneous maps of A 1 x M into E with multiplication defined by & l ' ( o l . P ) = C?Enl&(a, -;)il(-/, 0 ) . For k E K we define a llomogeiieous map k' of M x 2 1 1 into E by k l ( a , a ) = a k , k ' ( a , P ) = 0 if cu # B. Then X: --, k' is a homomorphism of K into B ( K ,1). This is a inollomorphism since ti' ~ X . FE = Eel '3 . . 8 Ee,, where the e , are orthogonal idenlpotent,~and for any k E K. k = C ( n i k ) e ,where nik E E. Then a , E M and if a,k = 0 for all i, k = 0. Thus we can identify I< with its image in B ( K ,1) and write k for k'. Then B ( K ,1 ) = K b y Theorem 2.5.9. We now consider the ma,p rl : C --i &' where E'(a!P ) = L(a,P)b(a,P ) for & t B ( K . 1). The condition (2.5.13) implies t,hat 7 is an automorphism of B ( K . 1). Moreover, (2.5.13) gives b(a,a)' = b(a.a ) SO b(a,a ) = 1. Hence qa = a for a E K . It follows from the Skolein-Noether theorem and B ( K ,l ) K = K that there exists an invertible a E K sucll that 71 = I,, the inner automorphism x --, aza-l. Now let v be defined by v ( a ,P ) = 1 for all a ;P. Then v is homogeneous and .c' = ' r ~ usatisfies c'(a. /3) = b(a,p). Since ( u v a p l ) ( a8) ! = (a(~)(,Oa)-l we have b(a,;?) = (aa)(Pa)-lfor a , P E M . :-

2.6. Crossed Products IVe shall now specialize to the case E = K . M = G = Gal E / F in the foregoing considerations. In this case one has the crossed product representation, due to Emrny Noether, of a central sirnple algebra A containing E and having dcgree n = [E : F].Let a E G. Then a can be extended to an inner automorphisrn I,," of A. B y Theorem 2.5.9. AE = E . Hence the element u , is determined up to a multiplier in E x . Moreover. since IUoIUTand IUOuT for 0. T E G hale the same restriction a7 to E , we have u,~L,= k,.u,,, k,, E E*.Also u,au;' = aa, a E E . Thus we have

2.6. Crossed Products

for a E E.

0.

r E G. The associativity (u,ur)up = u,(u,u,)

57

gives the relations

It is clear frorn (2.6.1) that tlie E-subspace Cute E u , is a subalgebra. On the other hand. it is easily seen by a Dedekind independence argument that the 11, are linearly independent over E. Hence [ C E u , : E] = IGI = n and hence [CEu, : F] = n L = [A : F].Thus

A

=CEu,

(2.6.3)

\Ve now consider the converse in which we begin with the Galois extension field E/F and the Galois group G. Then a rnap k of G x G into E * : (0.7) --i k,,, is called a Noetller factor set if (2.6.2) holds. We form the (left) vector space over E with base {2~,loE G) and we define a product in A = C E u , b y

Then (2.6.2) implies that this is associative. I\loreover. if we put a = r = 1 and T = p = 1 s~~ccessively in (2.6.2) we obtain

which imply that 1 = k:ul over F c E and we have

is the unit of A. LIoreover. A is a vector space

for x, y E A. a E F. Thus A is an algebra over F (associative with 1). This is called the crossed product of E wzfh G and Noether factor set k and is denoted as A = (E.G, k). The result we proved above can now be stated as

Theorem 2.6.7. I f A is a central simple algebra containing E and the degree o,f A i s n = [E : F] t h e n A is a crossed product (E,G, k). It is quite easy to prove directly the converse that any crossed product is central simple over F of degree n = [E : F]. We shall obtain this result by establishing the connection between crossed products and Brauer algebras. We note first that if we replace u l by 1 we may assume that the Noether factor set is nornlalzzed in the sense that kl,, = 1 = k,,l.is E G. Then we have

Theorem 2.6.8. If k is a normalized Noether factor set then c defined by c(p, a , r) = p k , - ~ , . , - ~ ~

is a reduced Brauer factor set and ( E ,G. k) reduced Brauer factor set then

-. B(E,c ) . Conversely,

(2.6.9)

if c is a

58

11. Brauer Factor Sets and Noether Factor Sets

defin,es a n,orm,alized Noether factor set and B ( E ,c) .- ( E ,G , k ) Proof. First, let k be a normalized Noether factor set. Then ( E .G. k ) = {Ca,u, / a, E E ) and we have

Now the element Ca,u, of ( E ,G , k ) can be identified with the map f : a --i a , of G into K. In this may ( E ,G , k ) = { f lG + E ) with t,he usual addition and multiplication by elements of F and with multiplication defined by

The unit of (E,G , k ) is now the map 1 such that 1 --t 1 and a --t 0 if a # 1. If k is a normalized Noether factor set then direct verification shows that c defined by (2.6.9) is a Brauer factor set, so we can define the Brauer algebra B ( E ,c) as the set of homogeneous maps of G x G into E with the usual addition and nlultiplication by elements of F and multiplication defined by (2.5.7).Now for f E ( E ,G , k ) we define [ f = !by

Then !(pa, pr) = p!(cr, 7 ) so [ f = !E B ( E ,c) and 6 : (E, G, k ) + B ( E ,c). Now [ is bijective since if we define 7 : B ( E ,c ) + ( E ,G , k ) b y ql = f where

then ( 7 0f = f for f E ( E .G. k ) and ([q)!= L for E E B ( E ,c ) . It is clear that [ is a vector space (over F) isomorphism. Now let f . g E ( E ,G. k ) . Then

and

Hence ([ f ) ( [ g ) = [f g and B ( E ,c).

6 is an algebra isomorphism of ( E !G , k ) onto

2.6. Crossed Products

59

Direct verification shows that if c is a Brauer factor set then k defined by (2.6.10) is a Noet,her factor set and the map c --t k is the inverse of the map k --, c defined before. Hence if we are given B ( E ,c) and we form (E,G. k ) where c --t k then ( E ,G , k ) -. B ( E ,c).' The Noether factor sets form a group under multiplication which contains the subgroup of factor sets of the form

-

where a --i L, is any map of G into E*. The factor group is the tohomology group H 2 ( G ,E " ) . As usual: wc write k 1 if k is in the subgroup defined by (2.6.13)and k k' if k and k1 differ by an elenlent of this subgroup. We now observe that the map k -, c is an isomorphism of the group of Noet,her factor sets onto the group of Brauer factor sets. If ?! is any map of G into E* then ! ( a ,r ) = o l ( o p l r )is a homogeneous map of G x G into E " . It follows that if k --, c in our homomorphism then k 1 if and only if c 1. Hence we have an induced isomorphism of H 2 ( G ,E * ) onto H 2 ( E / F ) .This isomorphism together with Theorem 2.6.8 permit us to carry over the results of Sections 2.4 and 2.5 to Noether factor sets and crossed products. We obtain in t,his way the classical result of Emmy Noether:

-

-

Theorem 2.6.16. Lef

E/F be

a

-

Galozs rxtenszon with group G . Thrn

-

(i) The crossed product ( E ,G , k ) 1 if and only if k 1 . (ii) ( E ,G , k l ) Ejz ( E ,G , k2) ( E ,G , klk2). ( ~ i i )Let [k] denote the element of H 2 ( G ,E * ) determined b y k. Then [k]--, [ ( E ,G , k ) ] is an isorr~orphismof H ~ ( GE,* ) onto B r ( E / F ) . T i c now assume E / F is cyclic. Let a be a generator of G and put u = u,. Then uZ( u , , ) ~ ' centralizes E ; hence uZ= e,u,, where e, E E*. We can replace u,, by u'. 0 5 7 5 n - 1. This replaces the factor set k by k' where

and y E E*.The crossed product A is generated by E and u arid every element of A can be written in one and only one way as

We have the relations ua = (aa)u, un = y

(2.6.19)

if a E E. Since lln commutes with u and with every element of E*. un is in the center F of A. Hence 3 E F*.Thus we see that A is the cyclic algebra

(E, o-.y?. The foregoing proof is more direct than the one given in Jacobson [831]. I am indebted to hlr. Guo Li for pointing it out to me.

60

11. Brauer Factor Sets and Noether Factor Sets

-

1 if and only if It is readily seen that if k' is defined by (2.6.10) then k' y = N ( ! ) for & E E * . This permits the specializatiorl of the rnain theorem on crossed products (2.6.16) to the following theorem on cyclic algebras. Theorem 2.6.20.

-

(i) The cyclic algebra ( E , c Ty) , 1 if and only if -1 = N E I ~ ( t for ) some P E Ex. (22) ( E . 0,fifl)8~( E , 0.72) ( E , O, 7172). (iii) Let N ( E * ) = { N E / F ( ! ) , !E E * ) . Tllen the map Y N ( E * )--t [ ( E O, , y)] is an isomorphism of F e / N ( E * ) with B r ( E / F ) .

-

The result just indicated shows that cyclic algebras can be regarded as special cases of crossed products. Fire shall now show that certain crossed products can be regarded as generalized cyclic algebras. We consider a crossed G, k) and we suppose H a G and G / H is cyclic. Let T be an product A = (E! element of G such that the coset r H generates G I H . Then any element of G has the form art where a E H and 0 5 i < r = IGIHI. hloreover, rr E H . Let D be the subalgebra of A generated by E and the u,, CT E H . If F' = Inv H then F' is the center of D. It is clear that D as algebra over F' is the crossed product ( E , H : kt) where k' = {k,,,, a . a' E H ) . Since H a G, rar-I E a' if a E H. Hence if we put u = u, then t,he inner aut,omorphism I, stabilizes D. Let 5 = &ID. Then we have the relation ua = (Ira)u, a E D. Moreover, uT = h ~ , r where !E E* and since rTE H we have uT = b E D . It follows that if D is a division algebra then A is a generalized cyclic algebra ( D ,Zr,b). In particular this is the case if A is a division algebra.

2.7. The Exponent of a Central Simple Algebra If A is a central simple algebra over F then A = hfT(D)where D is a central division algebra. If [D : F] = d2 then we have called d the zndex of A (or of [A]).Slre shall now prove that B r ( F ) is a torsion group by proving Theorem 2.7.1. If d is the index of A then [AId = 1.

Proof. Let E be a finite dimensional Galois splitting field for A. By replacing A by another element of [A] we may asslime that A = il/IT(D) = (E,G , k). Now A can be identified with Endno (V) where V is arl r-dimensional vector space over Do. Sirlce A contains the field E, V can also be regarded as a vector space over E. If [E : F] = n then [V : F] = [V : E][E: F] = n[V : El. Also [V : F] = [V : Do][DO: F] = rd2 and [ A : F] = n2 = r2d2 SO n = rd. Hence n[V : E] = nd and [V : E] = d. If {u,la E G) is the base for A = (E,G, k) such that (2.6.1) holds then the first of these relations shows that u, is a 0semi-linear transformation of VIE. Now let (xl, x2, . . . , xd) be a base for VIE and write

2.7. The Exponent of a Central Simple Algebra

61

Then Af" = (J\~,,(D))is the rnatrix of u, relative to ( x l , x2.. . . . xd) The T the niatrix relations relations u,uT = ,to T ~ ~ gimply

Let p, = det iLf". Then we have

-

-

Since u, is invertible. pC1, f 0 and herlce (2.7.4) gives k:, so kd 1. Hence Ad -- 1 by (2.6.9).

=,

U ~ ( ( T ~ and ,)~;~

The order of {A) in Br(F) is called the exponent e(A) of A. This is the e

1. The preceding smallest positive integer e such that A @ A @ . . . @ A theorem implies that the exponent is a factor of the index. We shall now show that these two integers have the same prime factors. Theorem 2.7.5. If p is a prime dividing the index of A then p divides the exponent of -4. Proof. Suppose p I d. We may assurne A = (E,G. k ) = n/f,(D) where [D : F] = d2. Tlien p I n = rd and hence p / /GI. G = Gal E I F . Let H be a Sylow p-subgroup of G, K the subfield of E such that Gal E I K = H. Then [E : K] = pn' and p 1. [K: F].Consider Ax. This is a central simple algebra over I( and since d f [K : F].K is not a splitting field for A so AK 1. On the other hand. E I K is a splitting field for AK so the index of AK has the form pS # 1. Since AK 1,e(AK) # 1 and since e(AK) 1 pS we see that p / e(AK). Since the map B --t BK is a homomorphism of B r ( F ) into Br(E) it follows that p 1 ?(A).

+

+

We shall now use the results on exponents to prove that any central dix-ision algebra is a tensor product of central division algebras of prime power degrees. We require the

Lemma 2.7.6. Let D l and D 2 be finite dimensional division algebras. Assume D L is cer~tralan,d ([Dl : F]:[D2 : F ] ) = 1. Then Dl~%r;.Dzis a division algebm. Proof. D l g F D 2 is simple (Corollary 2, p. 219 of BA 11).Hence Dl @ F D2 = ,VfT(D)where D is a division algebra. Now iWr(D) is a direct sum of r minimal . . . . I, and these are isomorphic modules for A/l,(D). Hence left ideals 11,12, they are also isomorphic as D,-modules. Then [D, : F] = [A& (D) : D,le for J # i is divisible by r . Since ([Dl : F],[D2 : F ] ) = 1 this implies that r = 1 D2 = D. and so D l @I;.

62

11. Brauer Factor Sets and Koether Factor Sets

Theorem 2.7.7. Let D be a central dzvzszon algebra of degrpe d = p t l . .p:,. -. Dl 8 D2 8 . . . @ D , where D , zs a central dzvzszon

11%dzstlnct przmes. T h e n D algebra of degree J I ~ ' .

Proof. Let e be the exponent of D. Then e = r l e a . . . e, where ei = p r ' ; 0 < ,In, 5 I;,. The cyclic group ( [ D l )has order e so [Dl = [ D l ][ D 2 ]. . . [D7.]where Di is a central division algebra of exponent e, and hence of degree I;: 2 rsr,. By 2.7.6. Dl 8, D2 8r;.. . . 8, D, is a division algebra and its degree is k' pl' . . . p $ . Hence D -. Dl 3 . . . % D, and the degree of D is . . . p $ . Thus I;, = k:. 15 i < s .

pt'

2.8. Central Division Algebras of Prescribed Exponent

and Degree \'lTehave seen in the last section that if D is a central division algebra of degree d and exponent e then (i) P 1 d (ii) every prime factor of d is a factor of e \Ve shall now show that if d and e are positive integers satisfying these conditions then there exists a D having degree d and exponent e . The construction we shall give is one given by Brauer in 1331. \'lTe assume first only condition (ii) on d arid e. Let P be a field of characteristic 0 such that 1. P contains a primitive e-th root 2. Ad - E is irreducible in P [ A ] .

E

of 1.

The following lemma gives an example of a field satisfying these conditions and gives some properties of any P satisfying the conditions. Lemma 2.8.1. ( 1 ) T h e cyclotomic field A, over Q of P - t h roots of 1 satisfies 1. and 2. (2) If P satis,fies the conditions 1. and 2. and p is a prime divisor of d t h e n P contains n o primitive pe-tlz root of 1. (3) If P satisfies the conditions 1. and 2. above and E is a n extension field of P that con,tains a n elem,ent 7 such that rid = E t h e n is a primitive de-th root of 1.

Proof. ( 1 ) VCTehave [A, : Q ] = $ ( e ) and [Ad, : Q ] = d(de). The condition (ii) implies that $ ( d e ) = d d ( e ) . Hence [Ad, : A,] = d. Let E be a primitive e-th root of 1, E' a primitive de-th root of 1. Since & I d is also a primitive e-th root of 1 we have A, = Q(eId)and A" is the niinirnum polynomial of E' over A,. Hence Ad - E'* is irreducible in &[A]. Since E and eld are primitive e - t h

2.8. Central Division Algebras of Prescribed Exponent and Degree

63

roots of 1 t,here is an automorphism of A, sending into E . Hence Ad - E is irreducible in A, [A]. (2) Suppose P satisfies the conditions 1. and 2. and P contains a primitive pe-th root E' of 1. Then P > A,, > A, and we have an autornorphism of 11, sending E'" into E . This can be extended t,o an aut,onlorphism of A,,. Since A" - E'P is reducible in Ap,[A] the same is true of XP - E . Hence A" - E is reducible in P[A]and, consequently, Ad - E is reducible in P[A] contrary to condition 2. (3) IVe have P > A, and E > A, (7).Ae (q) c Ad, and [A,(7) : Ae] = d = [Ade: A,]. Hence Ad, = Ae(q). Let E' be a primitive de-th root of 1. As before, we have an automorphism of A, sending E ' ~into E arld this can be extended to an automorphism of Ad, sending E' into 7. Hence 77 is a primitive de-th root of 1. Now let P satisfy 1. arld 2. and let E = P ( z l , . . . , z d ) the field of rational expressions in indeterminates x, over P. Let a be the automorphism of E/P permuting the x, cyclically. Let F = Inv(a) so E / F is cyclic with Galois group G = (0). Hence we can form the cyclic algebra A / F = ( E ,o,E ) where E is as in 1. and 2. We shall prove

Theorem 2.8.2. A = ( E , a, E) is a division algebra. For the proof uTeshall need some results on the action of 0 in the polynomial ring R = P [ z l . . . . , r d ] . For the present we drop the assumption on the existence of E and assume only that char P { d. We note first that if f E R t h e n N ( f ) = f ( a f ) . . . ( a d - ' f ) ~ S = F n R a n d i ff # O t h e n N ( f ) f Oaild f / N ( f ) in R. It follows that if f E E then there is a g # 0 in S such that gf E R. If t d we put

Then E = ~ ( ~ F1 = . ~ ( l )R, = ~ ( ~ S1 =, ~ ( l ) . Let V = Px,.Then V is stabilized by o so V is a P[a]-module. Since cr is a root of Ad - 1 and this polynomial is a product of distinct prime polynomlals, V = V , @ . . . e V, where the V , are irreducible P[a]-modules. Let A, E N and put YXI, = v': v:' . . ".:V c R. (2.8.4)

~f

,xT)

Then R is graded by the

,,..., A?):

R = @Yxl,....A,) (Al, . . . , A,) € N ( ~ ) Y A 1 ,...,~ ? ) Y f i..., 1 ,P T )

(2.8.5) ~~l+~l~...~~T+fir)

T.ZTeshall call this a o-grading of R. The elements of V(xl.,,,,xT) are said to be homogeneous of degree (A1,. . . ,A,). = Yx,,,,,,A,.).This implies that if t d then Evidently oV(x,..,.,

11. Brauer Factor Sets and Nocthcr Factor Sets

6d

This is equivalent to: if a E R ( ~and ) a = Ca(A,,,... A".) where a (,,..., ~ A".) t

y x ,,..., A,) t'hen a(/, ,,.... A,)

1% shall nerd the following

Lemma 2.8.7. Suppose t I d and t' / t and assume P cor~tainsa primitive d/tl-th root of unity. Then for any ( A 1 , .. . :A,) there exists a homogeneous element g f 0 i n R ( ~su.ch ) that (t) C~(t'), gV(~l,...~~T)

(2.8.8)

' condition that P corltains a primitive Proof. lye Iiwe ( ~ ~ ' )=~1l so~ the d/tf-th root of 1 implies that the characteristic roots of at' I V are contained in P. Since V, is an irreducible module it follows t,hat at' / V,c,lv3. Then at' I V ( X 1 , , . , = . ~c?' , ) . . . C:.- ~ v A , , .,A,.), Now the c, are d/tl-th roots of unity and they geriera,te the group of d/tl-th roots of unity. Otherwise, we have ( a I R)" = 1 for h < d and hence oh = 1 contrary to the fact that a has order d in E. Now let f t , , , , , x r ) . Then ot' f = c f , for c = c i l . . . c:., and = . . .c g ~ f = at f = ( a t ' ) t / "f = ctlL'f so ctlt' = 1. Also we have then at'g = c-lg and a t g = ( ~ ~ ' ) ~=l " g so if we choose g # 0 in ( c ~ ' ) ~ l= ~ 'g.g Thus g E R ( ~and ) a t ' ( gf ) = c-'cg f = g f . Hence g f E ~ ( ~ ' 1 . The argument shows that t,his holds for every f t l$tl,,,,,AT1.Hence me have

~(il! ,,,,,,,F)

(2.8.8). Proof of Theorem 2.8.2. Let I I , ~=

ti

E A = ( E ,a . E ) satisfy

( a a ) u , a E E, u d = E .

(2.8.9)

Then the elements of A can be written in one and only one way in the form a i u 2 ,ni E E . 1% have assumed that Ad - E is irreducible in P [ A ] .Hence "A E is irreducible in R [ A ] = P [ x l , . . . n:,, A] and hence in E [ A ] .A fortiori "A - 1s irreducible in F [ A ]so F [ u ]is a subfield of A. Since [F[ti]: F ] = d this is a maximal subfield and the centralizer A ~ I " ] = F [ u ] .B y Lemma 2.8.1 ( 3 ) , ,u is a primitive de-th root of unity. Now let t / d and consider the subfield F [ u t ] of F [ u ] . Since Ad - E is irreducible in F [ X ] (or in E [ A ] ) :Xdlt - E is irreducible in F [ A ]( E [ A ] ) since u t is a root of Xdit - E , ( F [ t i t ]: F ] = d l t . Also ut is a primitive delt-th root . the double centralizer theorem for central simple of 1. Let At = A ~ [ ' " ]By algebras (Theorem 4.10: p. 222 of BA 11): F [ I L is ~ ]the center of A t . It is c1ea.r from 2.8.9 that d-1

xi-'

'

At

=

{ C a j u J aji G E ( ~ ) } 0 t-1

(2.8.10)

2.8. Central Division Algebras of Prescribed Exponent and Degree

65

Now E ( ~jut] ) is a field since ut is a root of X d l t - E which is irreducible in E[X] and hence in E ( ~ ) [ X ] .The automorpliism Iu stabilizes ~ ( ~ ) and [ u ot~ = ] I, / E ( ~ ) [ restricts u~] to o on ~ ( ~Hence 1 . ct has order t and

as algebra over F [ut] . Now Al = F [ u ]and Ad = A. We shall now prove by induction on f that every At is a division algebra. Thus we assume every At(.t' < t , is a division algebra. Suppose At is not a division algebra. Then t > 1 and At corltairis zero divisors # 0.Let p be a prime divisor of t and put t' = tlp. TVe shall show that the existence of zero divisors # 0 in At implies the existence of such zero divisors in -4p. This will contradict the hypothesis on At! and prove the theorem. Now let a = n,lsi, b -b,?~' where ai, bi E E ( ~ ) [ usatisfy ~] a f 0. b # O.ab = 0. Since the ui,O i t - 1 are independent over E ( ' ) [ u tab ] , = O is equivalent to a system of polynomial eclliatioils in a, and otbi with coefficients in P[ut]and we are assuming that thcse are solvable for n, not all O and bi not all 0. We note also that if the a, and bi can be chosen in ~ ( ~ ' )then [ u we ~ ]shall have a # 0 , b # 0 in At, such that ab = 0. We now replace P by P[ut]in the field considerations at the beginning of this sect,ion. If we write P for P[u" Illen P coritains a primitive delt-th root of 1 and since t' = tip, delt = delt'p is a multiple of dlt' (since p I e by condition (ii)). Hence P contains a primitive d/tl-th root of 1. T4Te note that if we multiply the given a, by a suitable non-zero element of S we may assume . we may assume the b, E R ( ~ Next ) . we can express the a, E R ( ~ )Similarly the a, and b, as sums of homogeneous elements in the c-grading. Moreover, we can order the degrees lexicographically and thus regard N(') as an ordered monoid. Let (XI,. . . , A,) be the lowest degree of homogeneity of t,he non-zero homogeneous parts of all the a, and let ( p l , . . . , p,) have the same significance for the bi. Then it is clear t'hat if we replace each ai by its homogeneous part of degree (XI.. . . , A,) if there is one arid by 0 otherwise, and we make the same t,ype of replacement for the bi then the equations giving ab = 0 are satisfied. Thus we may assume the ai are homogeneous of the same degree (XI,. . . , A,) and the b, are llomogeneous of the same degree ( p l , .. . , p,). Since P contains a primitive d/tl-th root of 1 we can apply Lemma 2.8.7 to obtain ) that every ga, E R ( ~ ' Also ) . if me apply the an element g # 0 in R ( ~such lemma and observe that g in this lemma can be replaced by o k g for any k

~h-'

c;~'

< <

q!:,.., V ( t ) -

(since A - ( A I , . . ,A ~ and ) OR(^') = ~ ( ~ we ' 1 see ) that there exists an h # 0 in R ( ~such ) that bh E At/. We have ga # 0, bh # 0 and (ga)(bh) = 0 with ga, bh E At,. This completes the proof. TVe now assume both conditions (i) and (ii) on d and e and we prove

Theorem 2.8.12. A = ( E ,a , E ) has exponent e in Br(F) .

11. Bralier Factor Scts and Noether Factor Sets

66

Proof. Let e' be tlie smallest positive integer such that Et -

I , e'

I

= N ( E * ) .Since e. The assertion is equivalent, by the result noted at the end of

Section 2.7. to e1 = e. NOW suppose Eel = !\'?J/~(/L). LfTecan write xvhere f , g E R = P [ z l , .. . xd]. Then

.

f ( o f ) . . . (od-I f ) = ~ ~ ' ~ .(. (ad-Is). a ~ ) .

/L =

f/g

(2.8.13)

LVe assume deg f minimal. If deg f > 0 let q be an irreducible factor of f in R. Then y / o"q for some i and A i E / ~ ( 4 )1 A T E / ~ ( gin) R. \lie can caacel :YEIF(y) on both sides of (2.8.13) to obtain a relation (2.8.13) wit,h f of lower degree. Hence deg f = 0 and then deg g = 0. Thus eel = NEIF(h) with h E P and hence &=hd; ~ E P . (2.8.14) Tlle order of E"' in the multiplicative group P" is e/e1. On the other hand. the order of hd in P* is k/(d, k) where k is the order of h. Hence

The conditions (i) and (ii) on d and e and (2.8.15) imply that any prime dividing k divides e. Moreover, if k is divisible by a higher power of p than e then P " coritairis a primitive pe-th root of 1 contrary to Lemma 2.8.1 (2). it now folloxvs that k e. Hence k d and (d, k) = k . Then e I e, by (2.8.15). Thus e' = e.

2.9. Central Division Algebras of Degree

4.

LVe shall prove that these algebras are crossed product,^. The result for degree d = 2 is folklore. For degree three it is due to \tTedderburn and for degree four in the sharper form that any central division algebra of degree four contains cyclic of order k ) : it is a maximal subfield whose Galois group is Zz x Zz (Zk due to Albert [29]. rl = 2. The quickest way of obtaining the result for degree two is to invoke the theorem that such an algebra D contains a maximal separable subfield EIF (Theorem 1.6.19). Such a field is Galois. In a more elementary fashion the "difficult" case of characteristic 2 can be sett,led by the following argument. Let d E D be inseparable. Then d $4 F , d 2 E F . Choose a E D: $4 F ( d ) . Then b = [da] # 0 but [db] = [d[da]] = [d2a] = 0. Put c = ab-'d. Then = c2 1. Hence [dc] = d, dcd-' = c 1. Then dc2rl-' = (dcd-I)z = (c (I((:' c)cl-l = c2 c SO c2 c cornniutes with d and c. Since D is generated by cl and c. c2 c E F . Thus c2 c = y E F which evidently implies that c is a separable element and c @ F . d = 3. IVe shall give \ITedderburn's proof (Wedderburn [21])which is based on his factorization theorem for the minimum polynomial of an elenient of a central division algebra. This is the following

+

+

+ +

+

+

+

+

2.9. Central Division Algebras of Degree

5 3.

67

Theorem 2.9.1. Let D be a ,finite dimension,al central division algebra over a field F and let a E D and f ( A ) E F [ A ] be the m,in,imum polyn,omial of a over F . Suppose deg f = rn. T h e n we have the factorzzation

zn D [ A ] where a l = a and the a, ure co7~~7~gates of a . Moreover, zf u > 1 and we nzoy fake a2 = [ Y ~ ~ ] ~ I [ Y U . I ] - ~

6F

then

7n

(2.9.3)

where y is any element of D that does not commute with a . All but the last statement has beer1 proved in Corollary 1.3.14. TVe shall now give IATedderburn's proof of 2.9.1 including the last statement. This is based on

Lemma 2.9.4. Let D be a division ring and let a E D . Suppose ( A - a ) ,1 g ( A ) f ( A ) i n D [ A ] but ( A - a ) {, f ( A ) = boAm blAm-l . . . Om. T h e n R = b o a m + b l a m ~ l + . . ~ + b , # O a n , d ( A - R a n - ' ) I,g(A).

+

+

+

+

+

+...+

Proof. 1% have f ( A ) = Q ( A ) ( A- a ) R where R = boarn blarrL-I b, (see (1.3.10)).Since ( A - a ) i,f ( A ) , R # 0. N o w g ( A ) f ( A )= g ( A ) Q ( A ) ( A - a ) + g ( A ) R a n d since ( A - a ) ,1 g ( A )f ( A ) .( A - a ) ,1 g ( A ) R . Then ( A - R a R P 1 ) ,1 g ( A ) . \lJe can now give the

Proof of Theorem 2.9.1. The result is clear if al = a E F l . Now suppose a l 6 F 1 . Then m > 1 and there exists a y E D such that [ y a l ] # 0. Let y b r any such element of D. Sincc f ( a l ) = 0 we have f ( A ) = f I ( A ) ( A - a l ) and f ( A ) = y f ( A ) y - I = y f l ( A ) y - l ( A - y a l y - l ) . Since yaly-l # a 1 . R = y a l y - l - a l # 0 and by 2.9.4. A - R y a l y - l R - l ,1 f l ( A ) . Thus

where a2 = R y a ~ y - ~ Rwhere - ~ R = yaly-l Now suppose we have

-

0 1 . Hence a2 = [ y a l ] a l [ y a l ] - l .

where k < m . the a, are conjugates of a l and a2 = [ y a l ] a l [ y a l ] - lWe . claim t, fk ( A ) where fh ( A ) = ( A there is a conjugate a;+, of al such that (A-a;,,) a k ) . . . ( A - a2) (An - a l ) . Otherwise. we have a monic polynolnial fk ( A ) E D [A] of degree k < nl such that ( A - z a l z - l ) 1, f k ( A ) for all z # 0 . TVe may assume k minimal. Then ( A - a ) ,1 z-I f k ( A ) z for all z # 0 . This implies z-I f k ( A ) z = f k ( A ) for a11 z # 0 since if there is a zo # 0 such that z i l f h ( A ) z o # f k ( A ) then we have a monic polynomial g ( A ) of the form b ( z g l f k ( A ) z 0 - f k ( A ) ) of degree < k such that ( A - a ) ,1 z g ( A ) z p l for all z # 0. This contradicts the minimality of k . On the other hand, if z-l f k ( A ) z = f k ( A ) for all z # 0 then

11. Brauer Factor Sets and Noether Factor Sets

68

f k ( A ) E F [ A ] and since f k ( a l ) = 0 we have a contradiction t,o t h e hypothesis t h a t f ( A ) is t h e m i n i m u m polynomial o f a l . T h u s we have a conjugate aL+' o f a l such t h a t ( A { f k ( A ) . T h e n , b y t h e lemma, we have a conjugate ak+l o f a1 such that ( A - a k + l ) ,1 gk ( A ) . T h i s establishes t h e inductive step t h a t f ( A ) = g k + l ( A ) ( A- a k + l ) . . . ( A - a2)(A- a ' ) where t h e ai are conjugates o f al and a2 is as stated. W e shall call a n element a o f a central division algebra D cyclzc i f F ( a ) is a cyclic subfield o f D. T h e n me have Proposition 2.9.7. Let D be a central division algebra of prime degree p and let a E D have degree p. T h e n a is cyclic if and only if there exists c~ y E D such that yay-1 # a an,d [ y a y p 1 a] : = 0.

Proof. Suppose first we have a y satisfying t h e foregoing conditions. Since a is of degree p. P ( a ) is a maximal subfield, so t h e condition [ y a y - ' , a] = 0 implies t h a t yay-' E F ( a ) and hence a = Iy I F ( a ) is a n automorphisrn o f F ( a ) . Since yayp' # a: a # Since [ F ( a ): F ] is prime, F = I n v ( a ) and hence P ( a ) is Galois over F w i t h Gal F ( a ) / F = ( a ) . Conversely, suppose a is cyclic and Gal F ( a ) / F = ( a ) t h e n a a # a and we have a !/ E D such t,hat yayp' = cra. T h u s yay-' # a and [ y a y p l , u ]= 0. We can now prove t h e key

Lemma 2.9.8. Let D be a central division algebra of degree three o11er F and polynomial f ( A ) = let a be a non-cyclic element of D. Th,en the rn,ir~irn,l~nl A3 a 1 A 2 a a A of a ouer F h,as n factorization -

+

f ( A ) = ( A - Q ) ( A - a2)(A - a11

cu,c-'

=

(indices reduced mod 3 )

(2.9.9)

(2.9.11)

Proof. Sillcc f ( a l ) = 0 for a1 = a we have f ( A ) = g ( A ) ( A- n l ) . W e claim we can choose y E D so that [ [ y a l l a l ]# 0. Otherwise, z21 = 0 for t h e inner derivation ,,z = alL - a l ~ T. h e n a:LalR a l R = 0. Since al 4 F. [ F [ a l ]: F ] = 3 i o 1.(11 a: i11e linearly independent over F . T h e n t h e 9 linear transformations a4Lain 0 5 2 . 3 5 2 , are linearly independent2 T h i s contrad~cts a:L - 2 a l L a l R a f R = 0.

+

+

This is a special case of a general result on finite dimensional central simple algebras: If A is such an algebra over F and { a l : .. . , a , ) : { b l . . . . , b , ) are two sets of linearly independent elements, then the r s linear transformations i r, 1 j s : are linearly independent. (See e.g., the proof a , ~ b , ~1. of Theorem 4.6, p. 218 of Bh 11).

< <

< <

2.9. Central Division Algebras of Degree 5 4.

69

Now let y be a n element such that [ [ y a l l a l ] # 0 . Then yaly-' # a1 arid by IVedderburn's factorization theorem. f ( A ) = ( A - a s ) ( A - a 2 ) ( A- a l ) ~vhcrea2 = [ y a l ] n l[ ~ y a ~ ] Since - l . f ( A ) E F [ A ] it is clear that the factors of f ( A ) can be permuted cyclically. Hence ( A - a z ) 1, ,f ( A ) . On the other hand. ( A - a 2 ) ( A - a 2 ) ( A- a l ) . Otherwise. ( A - a 2 ) ( A- a l ) = ( A - b ) ( A - a 2 ) . Comparison of the coefficients of A shows that b = a l . Then ala2 = a z a l . Since a;! = R y a l y - l R - l and a1 is not cyclic. R y a l y p l R - I = a1 hy 2.9.7. Then [ R y ,all = O and since R y = yal - n l y . [ [ y a l l a l ]= 0 contrary to the choice of y Thus ( A - a2) i, ( A - a 2 ) ( A- a x ) . It now follows frorn Lemrna 2.9.4 that A - ( a z a l - ala2)a2(a2al - alaL)-l ,1 ( A - a s ) Hence

I,

Ncxt we use the relations

+

which come from A3 - a l A 2 a 2 A - a3 = ( A - a y ) ( A - n * ) ( A - a l ) = ( A a l ) ( A - a s ) ( A - a z ) = ( A - aZ)(A - a l ) ( A - a s ) . These imply

Now [ a l a z ]# 0 implies that ( A - a l ) {, ( A - a l ) ( A - a s ) . It follows as before that a2 = [nla3]al[ala3]-'. (2.9.16) Similarlv, we have the remaining formula in (2.9.11). By (2.9.11). ~ ~ a , = c -a,.~ 1 5 2 5 3. Since the a , generate D it follows that we have (2.9.12). MTecan now prove

Theorem 2.9.17. A n y central &vision algebra of degree three is cyclic. Proof. SVe have to prove the existence of a cyclic element not in F. Hence we begin with a non-cyclic element a and apply Lemma 2.9.8 to obtain an elerrlent c @ F such that c3 = 7 1 , 7/ E F. If this is cyclic we are done. Otherwise. we use this as the element a of the lemma and so we may assume that a3 = 01. a E F . Then we have A3 - a = ( A - a s ) ( A - aZ)(X - a l ) and ('2.9.10)-(2.9.12) hold. Now put bl = a l e , b2 = albla,' Then

Since a s a l a s = a = a:, azal = a; and

= a?ca,l.

(2.9.18)

70

11. Brauer Factor Sets and Noether Factor Sets

Hence alc2al = ca:c and [blb2]= 0 . If b2 = bl then. by (2.9.18). c = a l c a l l and a l c = cal. Since c a l c P 1 = a2 this implies a2 = a l and c = 0 b y (2.9.10). Thus ba # bl. This implies also that bl $ F and since [blbz]= 0 and b2 and hl are conjugates it follows frorn 2.9.7 that bl is a cyclic element $ F . The lnilii~nurnpolynomial of bl can he ralculated to be

For. we have

b: = a l c a l c = ca3ca3 = c2a2a3 (by (2.9.8)) =c

Hence

b:

2

(a3a2 - c ) = c2a3a2 - - .

+ ybl = ale3 0 3 0 2 = -,ala3a2

= YQ.

d = 4. The main structure theorem for central division algebras of degree -I can be stated in the following way: Theorem 2.9.21 A n y cen,tral divislon algebra of de,gree 4 that is a tensor produd of two separable quadratic fields.

contain,^

a slihfield

This is equivalent to: D is a crossed product (E; G. k ) where G Z2x 22. This result is due to Albert [29].Quite recently Rowen [78]has given a proof of the theorem that is constructive arld is similar to Wedderburn's proof in the degree 3 case.3 \;lie shall give a simplification of Rowen's proof which dispenses with the use of universal division a,lgebras and replaces this by more elerrientary Zariski topology arguments. \Ye shall first reduce the proof to showing that D contains a separable quadratic subfield. This reduction is achieved in the following two lemmas.

Lemma 2.9.22 Let D be a central division algebra over F , a a n element of D which is algebraic with min,irnum polynomial X 2 - aX - @ uihere a # 0 . Th.en there exists a n x E D such that y = [ a x ]# 0 and for swch a n .r we have ] 0: yay-1 = a1 - a and [ay2]= 0 . Hence F ( y 2 ) F ( y ) . [ay]= [ a [ a x ]#

5

Proof. Since a $ F1 there exists an x such that y = [ a x ] # 0. We have a2 = cva + /?. Hence a [ a r ] = [a21r] = a [ a z ]+ [axla. Thur a y + ya = a y and ya = (a1 - a ) y . If [ay] = 0 then a y = ( a 1 - a ) y so 2a = a l . Then 2y = [2a,r ] = [ a l ,x 0 so char F = 2. But then cvl = 2a = 0 contrary to a # 0 . O n the other hand, y2a = y ( a 1 - a ) y = a y 2 . Hence [ a y 2 ]= 0. Evidently [ay]# 0 and [ay2]= 0 + F ( y 2 ) F ( y ) . Also, yay-' = a1 - a is clear.

5

Lemma 2.9.23. Let D be a central di,uis,ion algebra of degree 4 and let F ( a ) be a separable quadratic szabfield of D . T h e n there exists a second separable It should be noted that Rowen's proof is similar to one given by Albert in thc characteristic zero case that was published in [322].

2.9. Central Division Algebras of Degree

5 4.

71

quadratic subfield Fib) such that the subalgebra F [ a ,b] generated by a and b i s the tensor product F ( a ) 8~ F i b ) . Proof. 'IVe show first that D coritairls an element x such that [ax]" F l . The set of these x's is an open subset in the Zariski topology. Hence it suffices to show that there is an x in n,f4(F)= D p $ F the algebraic closure of F , such that [ux]" F l . The condition on a irnplies that if uTereplace a by a siniilar matrix we nlay assume

This follou~sby elementary linear algebra. For. the minimum polynomial P,(X) is a quadratic polynomial irreducible in F [ X ]with distinct roots a l , a2 in F . Then a is similar in h f 4 ( F )to a diagonal matrix with diagonal entries a l , a2 where both a1 and a2 occur. Then the characteristic polynomial x,(X) = P , ( X ) U , ( X ) where u,(X) = ( A - a,)(X - a 7 ) ,z , 7 = 1 or 2. Since x,(X) E F [ X ] u,(X) . E F [ X ] .If u,(X) = ( A - a,)' then ( X - a , ) = (,u,(A). u,(A)) E F [ X ] . contrary to the irreducibility of P , ( X ) in F [ X ] . Hence v,(X) = P , ( X ) and (2.9.24) holds. Sl'e call write

Then

[ax]=

o (

a

-

a1)x21

(a1 - az)Xla 0

(2.9.26)

Then

[ax]' = -(a1

-

(2.9.27)

a 2 ) 2 d i a g { ~ ~ X21X12) ~~21,

and if we choose X 1 2 = e12.X a l = esl we shall have [axI4 @ F I . This proves the existence of x E D such that [axI4 $ F l . \Ve shall nou7 shoxv that if y = [ax]and y4 4 F 1 then b = y 2 is separable quadratic. [ab]= 0 and F [ a .b] = F [ a ]EF F [ b ] .By 2.9.22. F[b] F [ y ] .Hence F[b]= F 1 or IF[b] : F ] = 2. The first case is ruled out since b2 $ F I . Herice F[b] is quadratic over F and if this is not separable then b2 E F 1 agaln contradicting 7j4 $ F1. Also, by the proof of 2.9.22. [ab]= [ay2]= 0. Firlally [ F [ a b] , : F ] = 4 since otherwise a E F[b]and [ay]= 0 contrary to the relation = a 1 - a in 2.9.22. It remains to show that D contains an element a such that F ( a ) is separable quadratic. The main step in the proof of this is

5

Lemma 2.9.28 (Rowen). Let D be a central division algebra, a1 an element o f D having minirnum polynomial f ( A ) = X 4 a2X2- a 3 A a*. Then f ( A ) = (A2 - a'X b')(X2 - aA b) i n D[X] and for any such factorization. we have [ F ( a 2 ): F ] < 4 .

+

+

+

+

72

11. Brauer Factor Sets and Noether Factor Sets

Proof. The existence of the factorization into quadratic factors follows from \Vedderburri's factorization theorem (2.9.1).Also we have

and a2 =

a'a

+ b + b' = -a2 + b + n4bK1

(2.9.30)

We distinguish two cases: Case I [ah]= 0. Then a3 = (a4b-I - b)u and a: = [(n4b-l b)2 - 4a4]a2 = [ ( a 2 n 2 ) 2- 4n4]a2= n6 2a204 (a2 - 4n4)a2. Thus a' is a root of a cubic polynomial so [ F ( a 2 ): F ] < 4. Case I1 [ab] f 0. By (2.9.31), [a2b]= 0. Since [ab] # 0 it follows that F ( a 2 ) F ( a ) so again [ F ( a 2 ): F ] < 4.

+

+

+

+

5

To use Rowen's lemma to construct a separable quadratic subfield of D we begin with a pair of elements u , 11 and form a1 = [uv].Then the reduced trace t ( a l ) = 0. Suppose al has degree 4. Then the miniinusn polynomial m ( X ) of a1 over F has the form m ( X ) = X 4 n2X2- n3X C V ~and factors in D[X] as m(X) = ( A - a4)(X - a3)(X - aa)(X - ax) = (A2 - a'X b')(X2 - aX b) where b = anal. a = a1 a2. Also by Theorem 2.9.1. if u7e choose y so that [yal]= [y[ul;]] # 0 then we may assume that a2 = [ y a l ] a l [ y a l j K Then 1.

+

+

+

+

+

in Rowen's lemma. Now suppose we can choose u , v , y so that a* $! F . Then F ( a 2 ) is a separable quadratic subfield since. by Rowen's lemma. [ F ( a 2 ): F ] < 4 so [ F ( a 2 ): F ] = 2 or 1 and the latter is ruled out if a4 $! F1. Hence [ F ( a 2 ): F ] = 2 and F ( a 2 ) / Fis separable since otherwise the characteristic is 2 and (a2)' = a' E F contrary to the choice of u,v. y. TVe shall now prove

Lemma 2.9.33. If D is a central division algebra of degree 4 then D contains a separable quadratzc subfield. P ~ o o f .This will follow from the foregoing remarks if we can show that the subset T of D ( ~of) elements (u.v. y) such that 1. [u,V ] is of degree 4 2. [v, [uvll # 0 3. [[Y![uvl21[y,[ u ~ l l -$!~F) ~ is not vacuous. We can replace 2. by the polynomial condition

n the reduced norm; and 3. can be replaced by

2.9. Central Division Algebras of Degree

< 4.

73

where # denotes the reduced adjoint (x# = n ( x ) z p l if x is invertible). Let T I ,T2.T3 denote the sets defined by 1.2'. 3' respectively. It is clear that Tl and T2 arr open in ~ ( ~ Since 1 . the condition that z" I71 defines an open subset of D it is clear that T3 is open. Since the intersectioii of a finite nurnber of non - ~ ~ a c u o u open s subsets in the Zariski topology is non-vacuous open, it suffices to show that T I , 1 5 2 5 3, is non-vacuous and this will be the case if the corresponding subset T,of n f 4 ( F ) is non-vacuous. \TTeproceed to verify this. The proof of Lernma 2.9.23 shows that we may take

for any XtJ E Af2(F) and there exist such matrices having minimum polynomials of degree 4 (e.g. X12 = X21 = X w h ~ r eX 2 $ F12 and det X # 0). Thus TI # 0.We now take y = diag{Y, Y) where IT E -Z/I~(F).Then

so evidently

TZ

# @.Also we have [uv12=

For any matrix Z =

(11Zd2)l

diag{XlzXnl; X21Xlz) so

ZtI E h I 2 ( F ) ,wehave

n(z,,) = det Z i j . Hence

where

By (2.9.35) and (2.9.36)

where 212

=

[y, X ~ Z X Z I ] W I2 ~ 21

Hence z2 = diag{Z12Z211221Z12)and

=

[y,X21X1z]Wzl.

(2.9.41)

74

11. Brauer Factor Sets and Noether Factor Sets

where 6 = n([YXl2][YXZ1]). If we take

a simple calculation shows that

It is clear from this forrnula that the parameters a , p, a l . . . . can be chosen so that "z Fl. Hence T3 # GI. This completes the proof. Evidently Lemnlas 2.9.23 and 2.9.33 constitute a proof of Theorem 2.9.21. VTc shall now apply this theorem to obtain a canonical construction for central division algebras of degree 4 over F. By Theorem 2.9.21. D contains a subfield E / F that is abelian with Galois group V = (1. al, a 2 . 0 3 ) wherc a: = l , a z o 3 = a k . ? . J > k #. The field E has three quadratic subfields Q,, 1 5 7 5 3 , where Q, = Inv(a,). We have E = Q,Q3 -. Q , E F Q 3 for 2 # J . Let

Then [D, : F] = [D : F]/[Q, : F] = 8 and Q , is the center of D,. Evidently D L> E. The automorphism a, of E can be extended to an inner automorphis~n I,% of D . Since a, Q, = lQ2x, E D L and D, = E[x,]. Then D, is the cyclic algebra (or quaternion algebra)

over Q, where rc? = that

(1%

E Qi. The condition that Diis a division algebra is ai

@ N E I Q(bi) ,

(2.9.47)

for b, E E. Now let j # i. Since I x J Q i = Q,: I,. D, = D,. It is clear that D = D, [xj] = E [ z i ,x:,~]. Tie shall now make a normalization: We clloose x3 = ~ 1 x 2 ~vhichcall be done since a3 = alaz. Since the restriction of t,he automorphism IszIziIZ ; 1 to E is 02ala21= 01 we have 52x12i1 = axl.

We have D = Dl [x2]and if we write a = I,,

a E E*.

I Dl

(2.9.48)

then

and a2 = I,,. Hence it is clear that D is the generalized cyclic algebra R / R ( t 2 - a a ) . R the twisted polynomial ring D l [ &a ] . The coiidition that this generalized cyclic algebra is a division algebra is that

2.9. Central Division Algebras of Degree 5 1.

75

for y E D l (1.3.16). 55"e now derive some relations connecting the a, and a. We have (.cz;clz;1)2 =R : ~ x ? .= x ~a2al ~ and ( a ~ 1=) n~ z l a s ~ l = s ~n ( a l a ) a l . Hence: b y (2.9.48);

Similarly, since zlx2x,l

= a p 1 x 2 we have

Also we have a3 = :ci = (x1x2)" : ~ 1 : ~ 2 x 1=: ~. 2I . ~ ~ I I=: ~x lza;: ~ : ; ~ z ? ; c=; (a1a)alaa. Hence a = ((~10,3)(a1(alaz))-~ (2.9.53)

B y (2.9.51) and (2.9.53), we have (a2al)a;' = ( a l a s ) ( a l ( a l n a ) ) - l a 3 ( a l a a ) - l = ( g l a 3 ) a ~ 2 a 3 ( a 2 ( a 1 a 2 ) )and - 1 hence a3(01a3) = nl ( u a a l ) a z ( a l a z ) .Thus

\Ve can now provc Theorem 2.9.55 (cf. Albert [39], p. 186f). Let E be a quartic abelian extension of F with Galois groi~pV = { 1 , 0 1 , 0 2 ,a s ) such that a: = 1, aiaj = n k if i, j , k f . T h e n we have the Jbllowing recipe for constructing the central division algebras of degree fou,r over F ~ o n ~ t a i n i nEg: 1. Let Q1 = Irlv a1 and choose al E E such that al $! A ! E I Q ,( E ) . Form the quaternion algebra D l = ( E ,0 1 . a l ) over Q1. T h e n D l is a division algebra. 2. Let zl be a cnr~onicalgenerator of D l over E such that x l b = (a1b ) x l , b E E . Choose a E E such that (2.9.51) holds and a2 E E such that (2.9.52). Then! th,ere is an automorphism a of D 1 / F such th,at 0 x 1 = as1 and a 1 E = 0 2 . Moreover, a 2 = I a 2 . 3. Let R be the twisted polynomial ring D l [ t ;a a ] and D = R/R(t2- aa). T h e n D is central simple of degree four containing E and D is a division algebra if and only if (2.9.50) holds. on of degree 4 over F containing E can be Every central d % ~ ~ i s ialgebra obtained i n this way.

Proof. 1. This is clear. 2. We have the defining relations

in D l . If we put

2;

= (1x1 tllerl

76

11. Brauer Factor Sets and Koether Factor Sets

by (2.9.51). Hence wc have a n automorphism a of D l such that azl = x i and a / E = a1. Also a 2 = b; b E E. and a 2 x 1 = a ( a x l ) = (a2a)axl = a2(olcc,')rl: by (2.9.52). Hence 02x1 = I,,rl so o 2 = In,. 3. lye can form t,he generalized cyclic algebra D = R/kR(t" - 2 ) where R = D l [ t ; o ] . This is central simple of degree 4 (see section 1.4). By Theorem 1.3.16, D is a division algebra if and only if (2.9.51) holds. The fact that every central division algebra of degree 4 containing E is obt,ained following this procedure is clear frorn the analysis preceding 2.9.55.

2.10. Non-cyclic Division Algebras of Degree Four Albert has given a number of constructions of non-cyclic division algebras of degree four. His first construction was that of a tensor product of quaternion algebras. Later he gave two other constructions which are not tensor products of quatcrnion algebras, one containing an element a with minimurrl polynomial of the form X4 - CI and one containing no such element ([321],[33].[38r]).All of the ons st ructions are based on a result on cyclic quartic fields that we shall derive. For this we shall need the following norm theorem.

L e m m a 2.10.1 (Albert 1391). Let E/F be cyclic ,with Galols g3roupG = ( a ) o,f order r = r l r a . Suppose y is an elesment of F* such that 7'' = NE/F(c); c E E. Then there exists a cl E El = Inv(ar2) such that y = ATEIIF(~1). This can be proved quite easily using commutative methods. However, we prefer to give a lion-cornmutative proof of a more general result which we state as

L e m m a 2.10.1'. Let D be a division ring wzth an automorphism a such that ar = I and r is the order of u modulo inner automorphisms. Suppose r = rlr2 and 3 is a non-zero element of F = cent D n Inv(a) such that there exists a c scztisfyin,~yrl = Nr(c) = ( ~ ~ - ~ c ) ( o .~. .-c.~ Then c ) there exists (L cl E Inv(cr7.2)su,ch that 7 = IVr, (c) = (ar2p1c1)(ar2p2e1). . , el. Proof. Let R be the twisted polynomial ring D [ t ;a]. By Theorem 1.1.23. Cent R = F [ t r ] . Then t r is a two-sided irreducible element of R and (tr2- 7 ) / (t7- y r l ) . Since yrl = AT,(c), (t-c) (t7-:irl) (1.3.11). By Corollary 1.3.15. tT is a product of factors of degree 1. Hence the same is true of the factor trL - 7 of t r - ^fT1.Then 7 = ATr2(cl) = (aT2-1~1)(aT2-2c1) , . . c1 for some cl E D. Since a y = y we also have 7 = ( a r 2 c l ) ( a r z p 1 c l ).. . ( a c l ) = (oTzplcl). . . ( 0 ~ 1(ar2c1). ) Hence ar2cl = cl E Inv(ar2). We can now prove the following

2.10. Non-cyclic Division Algebras of Degree Four

77

a

Lemma 2.10.2. Let F be a field not containing (so char F # 2) and let E be a cyclic quartic extension field o~fF then the (un,ique) quadratic subfield K qf E/F has th,e form ~ ( d m?here ) u,v E F and u2 ti2 is not the square of an element of F.

+

Proof. Since char F # 2. K = F ( f i ) . w not a square in F . Now 1 = (-1)' = 1VEIF(1). Hence by 2.10.1, -1 = I?JKIF(cl), c1 E K . \Ire have cl = a+ b\fi, a.b E F.Then

Now b # 0 since nbK1, u = b-l.

aQ F. Hence b%l

= a2

+ 1 gives u!

= u2

+ u2,

u

=

Lelnlna 2.10.2 suggests a procedure for constructing a lion-cyclic division algebra of degree four: It suffices to construct a division algebra of degree 4 such that D 8~K is a division algebra for every quadratic cxtcnsion field K = F( JGi), PL. P' E F . For, then D contains no quadratic subfield of thc forin F( d G Z ) and hence, by 2.10.2, D contains no cyclic quartic subfield. ]Ye shall need a cor~ditiorlthat the tensor product of two quaternion division algebras is a division algebra. A first such condition is given in

Theorem 2.10.3 (Albert [72], Sah [72]). Let Di; i = 1,2, be a quaternion division algebra over the field F. Then Dl 8~ D2 is not a division algebra if and onlg if D l and D2 contaiir isomorphic quadmtic subfields. Proof. The condition is sufficient since the tensor product of isomorphic finite dimensional extension fields # F is never a field. Now suppose D l @ F D2 is not a. division algebra. WTeregard Dl and D2 as subalgebras of Dl D2 such that D l centralizes D2. Let Q be a separable quadratic subfield of D 2 and let m be the automorphism # 1 of Q I F . IVe have Q = F ( u ) where u2 = u a , cu E F'. a.nd cru = 1 - u. There exists a 2: E D2 such that

+

Suppose QD1 = Q 3~D L is not a division algebra. Then Q is a splitting field for D l and llerlce Q is isomorphic to a subfield of D l and Dl and D 2 have isomorphic quadratic subfields. Now suppose QD1 is a division algebra. Then it is readily seen that D I D z = Dl 8~ D 2 = QDl[u] is a generalized cyclic algebra R/R(t2 - 8)where R = QDl[t; u] and u I Dl = I D , uu = 1- u.Since D I D 2 is not a division algebra there exists a d E QD1 such that (ud)d = P (Theorem 1.3.16). We have rl = dl 21d2, di E D l , ad = dl (1 - u)d2 and the conditions p = (crd)d, u2 = u cu imply dldz = d2dl so Q' = F ( d l , d2) is a subfield of D l . Now consider Q 1 D 2 This contains Q'Q E Q' @ F Q. If this is Q and the result holds in this case. Now suppose Q'Q not a field then Q' is a field. Then Q'Da is the cyclic algebra (Q'Q, cr'. ,0)where a' I Q' = l Q / , a'u = 1 - u. We have /? = (ald)d for d = dl ud2 E Q'Q. Hence Q1D2 1

"

+

+

+

+

-

11. Brai~erFactor Sets and Noether Factor Sets

78

and Q' is a splitting field for D 2 . Then [Q' : F ] = 2 and Q' is isomorphic to a subfield of D2. Since Q' c Dl this proves the result in this case. [7

lye rio\x7assume char F # 2 and we shall obtain a quadratic form conditioi~ that the tensor product of two qt~aterilionalgebras over F is a division algebra. A quaternion algebra D, has a base (1.u,, u,. 1 1 ~ 1 - , ) over F such that

wherr a,!3? # 0. If both a , and 13, are squares then we ma) take these to be 1 and it is readily seen that D, M 2 ( F ) .If a , (or 3,) is a non-square then clearly D , is a cyclic algebra. In any case D, is central simple of degree 2. Let t , and n , he the reduced trace and norm respectively on D,. If T , = Eo El 11, + & v , <3u,u, then we have t ( x , ) = 2t0 and

"

+

+

+

+

The subspace Di = F u , Fv, Fu,v, is the set of elements of trace 0. On this subspace (2.10.6) reduces to

D, is a division algebra if and only if n , is anisotropic on Di. For. D, is central simple and hence D, is not a division algebra if and only if it contains an element J , # 0 such that x %= 0. For such an r , we have t , ( x , ) = 0 = n ( x , ) so n, is not anisotropic on Di. Conversely. if n , is not anisotropic on Di then wc have an x , # 0 with t ( x , ) = 0 = n ( x , ) . Then rp = 0. SVe now form Di 9D; and define a quadratic form on this vector space by (2.10.8) n ( x l xa) = n l ( x 1 ) - n z ( x a ) , 2%E DI.

+

Then we have the following result which is also due to Albert [321].

Theorem 2.10.9. Let D,, i = 1 , 2 , be a quaternion algebra over a geld F of characteristic # 2, D$ the subspace of Di of elements of reduced trace 0; ni the red~~ced norm on D,L.Define the quadratic form n on Di gjDh by (2.10.8). Then Dl @ F Dz is a division algebra if and only if n is anisotropic. Proof. If either Dl or D2 is not a division algebra then neither is Dl @ F D2. Moreover, since n , is not anisotropic on Dj for z = 1 or 2 it is clear that n is not anisotropic on D/1 @ D:. Now assume Dl and D2 are division algebra5 and Dl 8~ D2 1s not. Then, by 2.10.3. D, contains a quadratic subfield Q, n , ( x , ) l = 0 . Since Q1 r Q 2 we with Q1 r Q 2 . Now Q , = F ( x , ) where x: may suppose nl(z1) = nZ(x2).Then n(xl 1r2) = 0 and n is not anisotropic. Conversely, suppose we have x, E Di such that xl +xa # 0 and n ( x l +x2) = 0. Since D, is a division algebra. L , # 0 + n , ( x , ) # 0. Hence xl # 0, x i # 0 and n l ( x 1 ) = n 2 ( x 2 ) .It follows that if we put Q, = F ( r , ) then Q1 and Qa are isomorphic quadratic fields and hence D l @ F D2 is not a division algebra by Theorem 2.10.3.

+ +

2.10. Non-cyclic Division Algebras of Degree Four

79

a r e are now ready to define Albert,'~first class of examples of non-cyclic division algebras of degree 4. Let Fo be a real field (= subfield of R) and let F = Fo([, 7). [: 71 incleterminates. P u t S = Fo[[, 71. We order t,he monomials <9l,J lexicographically and if f ([; 17) E S and f (6: 7)f 0 then we define the leadirlg coeficler~t to he t,he coefficient of the highest iiionoliiial (in the lexicographic ordering) occurring in ~f(6; rl) and f (6. q ) is called monic if its leading coefficient is 1. If the highest rnonomial occurring in f ([. 17) is FZrlj then me call ((-1)'; (-1)J) the signat.u~re,sig f (E, 7 ) of f (6: 7). Let D, be the ~luat~ernion algebra over F with base (1; u,, r:i: IL, u,) such that (2.10.5) holds where cxl = 7; 0 2 = [. the ,8%# 0 E S; are monic and sig Dl = (1,- 1).sig p2 = (-1, - 1).For example: we may take pl = 7: P2 =
Theorem 2.10.10. D = Dl RF D2 is a non-cyclic division algebra. Proof. This will follow if we can show that for any quadratic extension field K = F ( 7 ) ; 7, = d m . a . P E F, DK is a division algebra. Observe that such quadratic extension fields exist. For example, E2 + 1 is not a square Now DK a division algebra imin F and hence we can t,ake 7 = plies D a division algebra and if DK is a division algebra for every quadratic extension K = F( d m ) then D contains no quadratic subfield of the form F ( d m ) and hence: by Lemma 2.10.2, D contains no cyclic quar= d m by p 7 , p # 0 in F and tic subfield. Since we can replace / I T = ,/I-~2(cx2 + D2) it suffices t o prove the result for -y = with a , 6 E S. Since ( D l E F D 2 ) N ~ D I K 9~D 2 it~ suffices to show that D I K '@K D 2 is~ a division algebra. This will follow from Theorem 2.10.9 if we can show that

d m .

-,

= 0 for

<,

E K only if every

Jw

<,= 0. We can write

and it suffices to prove the assertion for p z ,v, E S. We have

and since (I.-,) is a base for K / F . n ( t l , . . . , [6) = 0 implies that

+

Moreover, if Ei # 0 t>helieit>lierp, or u2 # 0 and hence Xi = p: u:-y2 # 0 since Fo is real. Thus it suffices t o show that (2.10.14) holds with the A, as in (2.10.15) only if every A, = 0. Since Fo is real a sum of squares of elements

SO

11. Brauer Factor Sets and Nocther Factor Sets

of S is either 0 or it has signature (1.1) and positive leading coefficient. It follows from the choice of and p2 that we have the following table: qAl = 0 or sig qA1 = (1, -1) P1Az = 0 or sip A2 = (1: - 1) 7701A3 = 0 or sig qP1X3 = ( 1 , l ) [A4 = 0 or sig [A4 = (-1: 1) 0 or sig P2A5 = (-1, -1) = 0 or sig [&AG = (1,- 1).

P2A5 =

In all the cases in which we have # 0 the leading coefficient is positive. Now suppose some A, for which (2.10.1i) holds is # 0. Let [ ' $ b e the highest monomial occurring arnoilg the terrrls listed above. We cannot have ((-1)J. ( - l ) k ) = (1. I ) , (-1.1) or (-1. -1) since the table shows that in these cases [ j V k occurs only in a single term in the list contrary to (2.10.14). Thus the only possibility is ((- 1)'. (- 1)') = ( 1 . 1 ) and [ z r l h c c u r s only in qA1, PIX2 and [P~AE.Since all of these terms have coefficients 1 ill the left hand side of (2.10.14) this equation holds only if every A, = 0.

2.11. A Criterion for Cyclicity of a Division Algebra of Prime Degree It is an open question whether or not all central division algebras of prime degree arc crossed products, or! equivalently, are cyclic algebras. For p = 2 and 3 this was proved in Section 2.9. For p 5 the question is open. If D = ( E , D, y) is of prime degree p then D contains an element u @ F such that up E F. We shall now prove that this necessary condition that a central division algebra of prime degree is cyclic is also sufficient. In fact, we shall prove a somewhat stronger result that given u such that u $! F, up E F ,t>henthere exists a cyclic subfield E / F of D / F of degree p such that u E u p l = E and D = I, / E is a generator of the automorphism group of E. Then D = ( E , 0, y ) and TL can be taken to be the canonical generator of D relative to E : u,a= ( ~ a ) uup, = y. We dispose first of the easy case in which char F = p. Suppose D is a central division algebra of degree p and characteristic p and D colltains an element u $! F such that up E F . Consider the derivation i, : x --, [ux]. We have i, # 0 and 2; = iUP= 0. Hence there exists a v in D such that i,v # 0 but i:v = 0. Put w =~(i~,v)~"u. (2.11.1)

>

Then since u and i,v are 2,-constarits. i,w = u. Thus uw

-

wu = u and

Put E = F(u1). Then (2.11.2) implies that uEu-I = E.Since [E : F] = p it follows that E I F is cyclic with 0 = I, I E as generator of Gal E I F .

2.11. A Criterion for Cyclicity of a Division Algebra of Prime Degree

81

StTe now assume char F # p and we proceed t o derive some results on c>clic fields o f degree p that we shall require. Let W = F ( [ ) where [ is a primitive p-th root o f unity. T h e n it is an elementary result o f Galois theory that TT'/F is cyclic and [LV : F ] = s I p - 1. Hence Gal bV/F = ( T ) where T ( < ) = Et. 0 < t < p. and s is tlie order o f t ( p ) in ( Z / ( p ) ) *W . e now consider an extension K/W o f the form W ( P & ) , a E W . and we prove the following iufficiellt condition that K is cyclic over F .

+

Lemma 2.11.3. Let PI' = F(E) where [ is a primitiue p-th root of 1 and let T be a generator of Gal W / F and T ( < ) = Let a be an element of W that is not a p-th power and ( r a ) a P t is a p-th power in W . Then K = W ( p & ) is cyclic over F of degree ps where s / p - 1 and K = TV @ F E where E is the unique subfield of degree p of K l F .

ct.

Proof. Put r. = P&. Since a is not a p-th power, [K : W ] = p and we have the automorphisin a o f K / W such that a ( r ) = [ r . T h e n o has order p. \Ve have ~ ( a=) bpat for b E W so ~ ( a=) (brt)p.It follows that the automorphism T o f I/V-/F can be extended t o an automorphism T o f K / F such that ~ ( r=) brt. Since a / W = l w and

a and T are comrnutirig elements o f Gal K / F . Since a has order p and T has order a multiple o f s (tlie order of .r / TV); (a:.r) contains an element 71 o f order sp. Since [ K : F ] = [ K : M,'][I..I~' : F ] = ps it follows that Gal K / F = (7). Hence K is cyclic o f degree ps over F and hence K contains a unique cyclic subfield E / F o f degree p. Evidently K = W @ F E.

-

-

Let s and t be as above. T h e n ( s , p ) = 1 = ( t , p ) so we havc integers s'. t' 5uch that ss' 1 (mod p) and tt' 1 (mod p). Now put

Then

and since t s

-

/

s

- -

\

S

t k t k = s' ( ? ( t t f ) * ) 1

1 (mod p), tIs

= 1 (mod p)

1 (mod p) and

t , = s't"

sf (mod p).

(2.11.7)

T h e following lemma gives a construction o f elements a E W satisfying the second condition: ( r n ) a P tis a p-th power in W . o f Lenima 2.11.3. Lemma 2.11.8. Let a E W " and put

Dl( a ) = l - I ( r k a ) " . 1

82

11. Brauer Factor Sets and Noether Factor Sets

T h e n ( ~ h ! f ( a ) ) h f ( a ) - is ' a p-th power i n W . Proof. Let IV*P be the subgroup of W * of p-th powers. If a . b E W * we write u =, b if aW*P = bliir'? We have

Since rs+' = T and t s r s' = t o (rriod p ) we have

On the other hand.

and since tt'

EE

1 (mod p)

Comparison of (2.11.10) and (2.11.11) shows that ( r M ( a ) ) l Z i r ( a ) 'E W * P . 17 We can now prove Theorem 2.11.12. Let D be a central division algebra of prime degree over F containing a n element u $ F such that U P E F . T h e n there exists a cyclic subfield E of D of prime degree over F such that u E u p l = E and a = Iu I E i s a generator of Gal E / F .

Proof (cf. Albert [382]).. The result has been proved if char F = p. Hence assume char F f p. As above, let W = F ( [ ) . [ a primitive p-th root of 1. Then p j [TY : F] and Dw is a division algebra. Now Dw contains the subfield K = F ( u ) 8~ W = I V ( u ) . If u p = 7 E F then K is the splitting field over F of XP - 7 . SVe have the automorphism a of I(/W such that o ( u ) = [ u . Then Gal K/W = ( a ) arld this is a normal subgroup of Gal K I F . Since K / W is cyclic with a as generator of the Galois group we have

Dw

=

( K , a, 6 ) .

6EW

(2.11.13)

as algebra over W . Then we have an element v E Dw such that va = (aa)v, a E K,

v P = 6 E IV.

(2.11.14)

ct

We know also that W / F is cyclic with Gal W / F = ( T ) where T ( [ ) = and [VV : F] = s where s is the order o f t + ( p ) in ( Z / ( p ) ) * .The automorphism r has a unique extension to an automorphism T of Dw = 14f % F D which is the identity on D. As a special case of (2.1 1.14) we have

2.11. A Criteriorl for Cyclicity of a Division .Algebra of Prime Degree L'U

Applying

T

Since

= Etuot we obtain

!?"I

= €7~1).

83

(2.11.15)

t,o this we obtain. since r ( u ) = u:

Then ~ ( 6 = ) r ( v P ) = ( a l u t ) ~ o ~ ( o t a l ) ( o Z t.a. l. )( ~ ( p ) ~ a ~ Since ) z ~ ~ (p, p .t) = 1 , Gal K I I V = ( L T ~ )and hence we have

If we apply r to this and note that T E Gal K / F and ) S ~ ~of . this gives obtain ~ ' ( 6 =) N ~ ~ ~ ( U ZIteration

(LT)

a Gal K I F we

Now define t k as in (2.11.5). Then, by (2.11.19),

Since. by (2.11.6). ztktk=. 1 (mod p) and [ K : W] = p we see that S and M ( S ) differ by the norm of an element of K . It follows that we can replace 2% by an element u l = bv. b E K, and obtain 7ua = ( a a ) u l .n E K , u,P = A f ( 6 ) . B y Lemma 2.11.8. A f ( 6 ) E W has the property that ( T M ( S ) ) M ( S ) - is ~ a p-th power in Mr. hIoreover, since DU? 1 , M ( 6 j is not a p-th power in W . Hence, by Lemma 2.11.3. W ( w ) contains a unique cyclic subfield E / F of degree p. Since u,= bv,b E K. we have from (2.11.15). that

+

Since W ( w ) / F is cyclic of degree sp and [W : F ] = s we have W (w) = W g FE. Since I,, 1 IV = l w it follows from (2.11.21) that ! L - ~ W ( W= ) UW ( w ) and since E is the only subfield of degree p of W ( w ) we see that I, I E is an automorphism p of E I F s~ichthat Gal E I F = ( p ) . It follows that E and u generate an F-subalgebra of Dcv that is a cyclic algebra (E,p,n/). Then ( E .p, 7).Then Dw = T.V @ F D = W

Since the degree of D @ F ( E , p ,y - l ) is p2 and [W7: F] = s it follows that D g F ( E . p, - I ) N 1 in Br ( F ) . Then D E ( E ,p, Y). This isomorphism implies that we have an element u' E D such that u'P = y and a cyclic subfield E' such that I,, / E' is an automorphism generating Gal E ' I F . Then F ( u l ) F ( u ) under an automorphism such that u' --i u.This isornorphism can be extended

"

11. Brauer Factor Sets and Nocther Factor Sets

84

t o a n inner automorphism of D. Tlle iinage of E' under this automorphism is a field EIF satisfying the conditions of the theorem. i7

2.12. Central Division Algebras of Degree Five We shall now apply the cyclicity result of tlle last section to derive a result of Brauer's ([38]) on splitting fields of central division algebras of degree five. Let D be a ccntral division algebra of degree n over F.K = F ( u ) a maximal separable subfield of D : f (A) t,he minimum polynonlial of u . E = F ( , r l , . . . , T,) a splitting field of f (A) where f (A) = 17(A - r i ) . As we have seen in Theorem 2.3.17 and its proof, we can identify D with the F-subalgebra of M n ( E ) of matrices of the form (kzSczj)where v = (cv) is fixed with every ciS # 0 and L = (I;,,) satisfies the conjugacy conditions (2.3.5). Since D E = Anfn(E) the characteristic polynomial of the matrix (!,,cij) E D is the reduced characteristic polynomial of this element of D and hence its coefficients are contained in F . This polyiiomial is

where hi, is the sum of the principal minors of rank k of (!,,7cz,). Now let g(X) = a0 alX . . . an-lA7L-1 # 0 for a, E F and define tZJby

+

+ +

!,

= 0, l,, = g(r,)-' for i

# ,J.

(2.12.2)

Then these satisfy the conjugacy conditions and (I;,,cL3) E D. We shall now derive a set of conditions on the ai to insure t,hat = ... - hnPl = 0 arld llerice that the reduced characteristic polyrlornial of t,he element (l,,c,,) reduces t o ATL (- 11, h n . For this purpose we introduce n indet,erminates [i. Then D = D F ( E l , , , , , t n ) over = F(t1? . . . , FrL)is a central division algebra and K = F ( u ) is a maximal subfield of D (Proposit,ion 1.9.1). W e have the splitting field E = F ( r l , . . . , rn] of f ( X ) . We can regard D as the set of matrices (?zjczJ)where the ii, E E satisfy tlle conjugacy conditions. Tlle characteristic polynomial i of such a matrix has coefficients in F . Now choose &, = O, FLSij~ri)pl = ([0 t1r, . . [n_lrr-l)pl for i # j . This gives an element of D whose chara.cteristic polynomial is %(A) = An - i%lA:pl . . . (-l)nhn,. Since hk is tlle sum of the principal minors of rank k of (li,cij) it is clear that if we put

+

+

+

+

+ +

.

Then P,-k = PnZk ( t o ,. . . [, 1) is a homogeneous polynomial of degree n - k in the ['s. Since hk and 17G(r,) E F the coefficients of Pn-k(Eo,. . . , Enpl) are contained in F. Also since l,, = 0, hl = 0 and hence Pn-l([o.. . . , SnP1) = 0. It is clear that if the ak E F satisfy

2.12. Central Divisiorl Algebras of Degree Five

85

and ( n l , . . . , a,-1) # 0 = (0, . . . : 0) then the corresponding element of D satisfies a pure equation An (-l)nh, = 0. r\/Ioreover,since the E,, = 0 it is clear that the element is not in F. If n is a prime it will follow from Theorem 2.11.2 that D is cyclic. Now let n = 5 a,nd, for t,he sake of simplicity, assume char F # 2. In t,his case we have the three conditions P3( a o ,. . . , a d ) = P2( a o ,. . . , a 4 ) = Pl(ao.. . . , a d ) = 0 where P k ( t o , . . , E4) is a homogeneous polynomial of degree k. Now PI = 0 defines a hyperplane. Hence the determination of the n , satisfying P3 = Pz = PI = 0 amounts to determining a point of intersection of a quadric arid a cubic surface in projective four space. While such an intersection may not exist for the base field F we claim that it does exist in an extension field obtained by adjoining two square roots of elements of F and then the root of a cubic equation. To see this we note that we may assume the 4 quadric is given by P2 = El crlx:. Then it is readily seen that if we adjoin JG, JG to F we obtain a line on P2.To obtain a point of intersection of P2 with the cubic surface P3 = 0 it suffices to obtain an intersection of this line with P3 = 0. This can be dolie if the field is extended by a root of a cubic equation. Mk therefore have the following

+

Theorem 2.12.5 (Brauer [38]). Let D be o, cen,tral division algebra of degree five over F (char F # 2 ) . T h e n there exists a field K of the form F ( & , fi.8 ) where a . p E F and Q is a root of o, cubic equation over F(@, such that DIc is cyclic.

a)

This shows also that D has a splitting field E such that E contains a subfield K over which E is cyclic of degree five and K is as in the theorem. If char F # 2 , 3 then the normal closure of E is solvable, that is, is Galois with solvable Galois group. Hence we have Corollary 2.12.6. A n y cen,tml division algebra of degree five has a solvable splitting field.

Note. Rosset has shown in [77]that if F contains p distinct p-th roots of 1 then any central division algebra of degree p over F has an abelian splitting field. This implies that any central division algebra of degrec p over a field of characteristic # p has a solvable splitting field of a very simple type. An extension of this result that is a. consequence of an important theorem of I\lerkurjev and Slislin will be proved by Saltman.

86

11. Brauer Factor Sets and h-oethcr Factor Scts

2.13. Inflation and Restriction for Crossed Products Tl'e now resume our study of the Brauer groups Br(F) and B r ( E j F ) where E is finite dimensional Galois over F. We derive first two preliminary results on semi-linear trarlsformations of a vector space.

Lemma 2.13.1. Let V be a vector space over the finite dimensional Galois extension ,field EjF. Suppose for each a E G we have a a-semi-linear transformation, PL, of V (u,(ax) = (aa)u,x) such that

Let Vo = {(YE 'b / u,y = y, a E G ) . Then Vo is an F-subspace of V and the canonical map n @ y w ay of VOE= E tit^ VO into V is an isomorphism. Proof. The assertion amounts to the following: V = EVo and elements of Vo that are F-independent are E-independent. That Vo is an F-subspace is clear. It is clear also that for any x E V. y = Eu,x E Vo. Now let ( b l , . . . b,) be a base for EIF. Then the elements

.

Now the matrix (a,bi),G = { a l ; . . ..a,), 1 5 i 5 n.: is invertible ( B A I, p. 292). Hence we car1 solve the system (2.13.3) for the u,x and express these as E-linear combinations of the y, E Vo. In particular, since u1 = 1; x is an E-linear combination of y, E Vo. Evidently this implies that V = EVo. Kext suppose yl, . . . ; y, E Vo are F-independent. Then the standard Dedekincl independence argument shows that these elements are E-independent. O If V is a finite dimensional vector space then Lemma 2.13.1 implies (and is equivalent to) a classical result on matrices due to Speiser [19].This is

Lemma 2.13.4. Let E/F be Galois with Galois group G and let a a ,map of G into GL,,(E) such that

--i

hI, be

Then there exists an N E GL,,(E) such that

Proof. Let u, be the a-semilinear transformation of an m dimensional vector space V I E having the rnatrix &Io relative to a base ( x l , .. . , x,) for V I E : V,,IC, = p,,,x, where h!I, = (p,,,). Then (2.13.5) implies that U , U , = u,,. Also the fact that the 12.1, E GL,(E) implies that the ZL, are bijective and hence 711 = u1u1 implies u1 = 1. Thus we can apply Lemma 2.13.1 to obtain a base (y,. . . . . y,,) for V I E such that the y, E Vo. Hence u,y, = y,, 1 5 z nL,

x3

<

2.13. Inflation and Restriction for Crossed Products

87

and so t,lle matrix of 7~, relative to ( y l : . . . , ym) is the identity matrix. Then if N is the matrix expressing the y's in terms of the x's: y, = Cu3zx3, N = (uL3) we have 1 = N-lA.l,(ufV). Hence (2.13.6) holds. We suppose next that V and V' are vector spaces over E and 7~ and IL' are a-semilinear transformations of V and V' respectively. Consider V E E V'. The map x @ x' --i ux 8 u'x for x E V, x' E V' is additive in both arguments and for a E E, u(ax) @ u'x' = (aa)ux @ u'x' = ux @ (ua)u'xl = ux 8 u1(ax') Hence we have a balanced product of V and V' and so we have a unique endomorphism u u' of the additive group of V @E V' such that 8%

(u 8 ul)(x @ x') = uz @ u'x.

(2.13.7)

This is a-semilinear, since if a E E, then

x')) = (IL 2 u') (ax 9x') ( ~ 8 '(a(x ) = u(ax) 8 u'x' = (aa)ux 8 u'x' = a a ( u x 8 u'x'). SVe shall now apply the foregoing results to the follou~ingproblem. Let E / F be Galois with Gal EIF = G and suppose E is a subfield of EIF that is Galois with Gal EIF = G. Then we know that H = Gal EIE 4 G and the restriction map u --i 8 = a 1 E is a homomorphism of G onto G with kernel H so G 2 G I H . Suppose we are given a crossed product A = ( E .G, k). Then A is split by E arld hence by E. Accordingly. A is similar to a crossed product (E.G. k ) . What is the relation between k and k? This is given in the following theorein which is due to Hasse (1331).

Theorem 2.13.8 (Inflation Theorem). Let EIF be finite dimensional Galois, EIF n Galois subfield, G = Gal EIF, G = Gal E/F,a * a = a I E the canonical homo~norphisrr~ o f G onto G. Let ( E ,G, k) be a crossed product of G with factor set k. T h e n k

-

:

(a, T ) * k,,, = k,,,

(2.13.9)

is a ,factor set of G with values i n E* and (E,G,k)

-

( E , G , k).

(2.13.10)

Proof. 4 s in the proof of Theorem 2.7.1, we can identify A = (E,G, k) with EndDoVwhere V is an I- dimensional vector space over a division algebra Do and [V : E] = rn, the index of A ([D : F] = m2). Let u,, u,, u , 7 E G, be elements of such that

88

11. Brauer Factor Sets and Kocther Factor Sets

Si E E. Then u, is a 5-semilinear transformation of V over E . Also a is a 8-semilinear transformation of EIE since a(tia) = (a(tia) = ( a a ) ( a a ) = ( ~ ? i ) ( a afor ) ti E E. a E E. Hence we have a a-semilinear transformation u, of V = E BE v over E such that

a E E, rt- E V. Moreover. since a ii also a a-sernilinear transformation of E it follows from (2.13.12) that u, is a a-semilinear transformation of V. By (2.13.12) wc hasre U,U, = k,,r~,,. (2.13.13) -

It follows that k defined by (2.13.9) that of V. By (2.13.12) we have

11,

is a a-semilinear transformation

It follows that k defined by (2.13.9) is a G-factor set with values in E* and A = COEG Ezi, ( E , G, k). It remains to show that ( ~ n d F v ) * ( E ~ ~ F since V ) ~these algebras are similar to A' and A' respectively (Theorem 4.11, p. 224 of BA 11). Since A = CEu, it is clear that ( ~ n d C ~L ~= )EndEV~ and ( ~ n d =~ ) *{ P E EndgV I {! E EndEV I 2~,&u;' = Il, a E G). Similarly ( ~ n d ~ V = ~ L ~ & L ;=~ k : 8 E G). Now a, : & --+ u , l ~ ; ~is a a-semilinear transformation ) ! by Lemma 2.13.1. L = of L and a1 = l L .Hence if B = ( ~ n d ~ ~then EB % E @E: B. Similarly, if B = (EndFV)* arld L = EndgV then L = EB E B. By definition, V = E 8~ V. Hence identifying V with the corresponding subset 1 8 V of V : any ? E L has a unique extension to a linear transformation of V I E which we shall also denote as 2.In this way we can E 8,q 1.Hence if (ll,. . . , i,,e,,z) is regard L as a subset of L. Then L = EL a base for L I E then this is also a base for L I E . Since L = EB r E @F B we may assume that 2%E B. Then any element of L can be writ,ten in one and only one way as Il = C a i & ,az E E and every element of L has this form with the ai E F. The condition a,! = !for E = ,En,& is equivalent to an, = ai, 1 5 i 5 m2. Hence B = B or, more precisely, B is the set of ext,ensions t,o linear transformations in VIE of the linear transformations of VIE E B. Hence ( E ~ ~ F V ) * ( ~ n d ~ a.nd ~ ) (' E ~ ~ F v ) " ( ~ n d p V ) ' as required.

-

"

"

"

-

The crossed product A = (E.G. k) defined by A = ( E , G. k ) is called the znflatzon , Infg,EA. An important application of inflation is the following result due to Brauer ([32]).

-

Theorem 2.13.14. Let A be central simple with [Ale = 1 where e is not divisible by char F . T h e n A ( E ,G , k) where the k,~, are e-th roots of unity.

-

Proof. We may assume A = ( E . G, k ) where EIF is -Galois with G = Gal E I F . - Since Ae 1 we have ic E E such that &,: = t,(5!,)l;: . Since e is not

~

)

~

2.13. Inflation and Restriction for Crossed Products

89

z,

divisible bv char E the polynomials A' are separable. Hence there exists an exterlsion field E / E such that E / F is finite dimensional Galois and E contains an e-th root E, of &. Let ~ n f ~ , ~ G. ( Ek .) where G = Gal E / F and k is as defined before. Consider

Then E: ,= 1 and ( E ,G. k ) = ( E ,G. E ) , ( E .G. k ) ( E .G ,E ) .

-

E

= {E, .). By the inflation theorem.

We investigate next the behavior of a crossed product under extension

of the base field. Let E / F be Galois and let F' be any extension field of F (possibly infinite dimensional). Since E is a splitting field over F of a separable polynomial f ( A ) E F[A],the splitting field E ' / F 1 o f f (A) contains F' and E as subfields. LIoreover. E' = EF' and E' is Galois over F'. It is readily seen that up to isomorphism over F there is only one extension field E' of F containing E and F' as subfields and generated by E and F'. SfTecall E' the composzte of E / F and F'IF. Let F" = E n F'.

Let G' = Gal E1/F'.H = Gal E/F1'. If a' E G' then a' / E E H and the map a' --, a' I E is a homomorphism 77 of G' into H. We claim that this is an isomorphism. First, it is injective since ~ ( a '=) 1 implies that a' E = 1~ as well as n' F' = IF,. Then a' I E' = 1 since E' = EF'. Thus a' = 1. Next 17 is surjective. Otherwise Inv q(G1)2 El' whereas Inv q(G1)= Inv G' n E = F ' n E = E". Hence r/ is an isomorphism. Then [E' : F'] = G' = H = [ E : F"]. This implies that

E'

= EF' 2

E E F 8 F'. ,

90

11. Brauer Factor Sets and Noether Factor Sets

We can now prove the Theorem 2.13.16 (Restriction Theorem, Hasse [33]). Let E I F be finite dimensional Galois with Galois group G and let F' be a n extension field of F, E' the composite of E and F', G' = Gal E'IF'. T h e n for any factor set k of G into E* we have (2.13.17) ( E .G , k ) ~--' ( E l ,G', k') where k;,,,, = ~ , , E , ~ , a', E ,T' E G'. (2.13.18)

Proof. Let the notations be as above and put A = ( E ,G , k ) . Then AFi = ( A F , / ) F ,Since . F" is a subfield of E . hence of A , by Theorem 4.11 of BA I1 (p 224), AF" -- AF".The latter has center F'' and a simple calculation shows that if u,, a E G , are the canonical generators for A over E then ilF" is the subalgebra generated by E and the u,, a E H . It follows that AF" 2 ( E ,H , k H ) where kH is the restriction of k to H . Then Ap, -- ( E .H , k ~ ) . Since E' = EF' E E @ F J I F' it is clear that ( E ,H , k ~ ) 2~( El l ,GI, k') where k' is given by (2.13.18).Hence A p -- (E',G', k'). The factor set k' is called the restriction of k a,nd we have the restriction hornornorph,ism R,es : [k]-, [k'] of H 2 ( G ,E * ) into H2(G',El*) . We have the following comnlutative diagram

H 2 ( G ,E * )

--+

Br(E/F)

where the horizontal maps are the isomorphisms [k] --i [ ( E .G. k ) ] and [k']--, (El.G', k ' ) , the left vertical is Res and the right vertical is [ ( E G . , k ) ]--i [(E, G. k ) ~ ' ] . The two re5ults we have derived spccialize easily to the following results on cyclic algebras which we state without proofs. Corollary 2.13.20. Let E I F be cyclic 'with Gal E I F = ( a ) and [ E : F ] = n. Let E be the intermediate field with [ E : E] = m and let d = 0 I E . T h e n ( E ,a, y) -- ( E ,a, ynL) ( E ( E ,a, 7)@ . . . X ( E ,a:7 ) :m timws). Corollary 2.13.21. Let E I F be cgclic with Ga,l E I F = ( a ) and let F' be a n extension field of F . Suppose E' is the composite of E and F', [E' : F'] = m; and a' is the extension of anlm t o E 1 / F ' . T h e n ( E ,a, y ) p -- ( E l ,D', 7).

We remark that Albert's norm theorem (Lemma 2.10.1) is an immediate consequence of 2.13.20 and the theorem that ( E .a, y ) -- 1 if and only if 7 = ILTEIF(u) for some u E E .

2.14. Isomorphism of B r ( F ) and H 2 ( ~ ) \.ZTe need to develop first some general results on the cohomology of groups. Let H and G be groups. a a homomorphism of H into G. Then any G-module A (BA 11. sec. 6.9) becomes an H-module via a by defining the action of H on A by (ah)x. h E H, x E A. Now suppose B is any H-module. Then a map s of A into B will be called compatzble wzth a if it is a module homomorphism of A as H-module into the H-module B . The condition for this is that for any z E A arid any h E H we have

Observe that if a is bijective then this can be written also as s(gx) = (a-lg)sx which is a generalization of the definition of a-l-semilinear transformation of one vector space into a second one. Now let f E C n ( G ,A) the additive group of n-cochains with values in A. We can associate with f and the map s (compatible with a ) an n-cochain S f of H with values in B defined by Sf ( h l , . . . , h,) = sf(ah1, . . . , ah,), h, E H.

(2.14.2)

Evidently S : f --i S f is a homomorphism of the additive group Cn(G,A) into C7"H, B). Moreover, this commutes with the coboundary operator f -i Sf where 6f E C n + l ( G ,A) is defined by

The commutativity means that we have the commutative diagram:

This follows directly from the definitions. As a consequence of this commutativity, we have an induced homomorphism of the cohomology group H n ( G ,A) = Zn(G. A)/Bn(G, A) into H n ( H ,B) (BA 11, loc. cit.). One important special case of these considerations is that in which H is a subgroup of G. a is the injection of H into G. A is a G-module and A is regarded as H-module via the injection 2 . Then the identity map is trivially an H-homomorphism of A as H-module with A as H-module. The corresponding homomorphism of Cn (G, A) illto C n (G, A) maps f E C n (G. A) into if where i f (hl , . . . . h,) = f ( h l , . . . , h,). The corresponding homomorphism of H n ( G ,A) into H n ( H ,A) is called the restrzctzon homomorphzsm.

92

11. Brauer Factor Sets and Noether Factor Sets

Of particular interest for us is the case in which U and V are normal subgroups of G and U > V. We have the canonical homomorphism gV --i gU of G/V into G/U. Let A be a G-module and let AU(AV)be the subset of A of elements x such that ux = x.u E U (ux = x,u E V). Then AU and AV arc submodules since U 4 G and these can be regarded in the natural way as G/U and G / V modules respectively. Evidently AU C A" so we have the injection homomorphism of AU into A" (as additive groups). If x E AU and g E G then (2.14.5) (gV)x = gx = (gU)x which shows that the injection of A" into A" is compatible with the hornomorphisrn of G/V into G/U. Hence we have the corresponding homomorphism. called the znflatson inf(U, V), of Hn(G/U, AU) -+ Hn(G/V,A"). This maps the cohomology class f + B(G/U, AU) into the class of the cocycle fLnf (U, V) where f Z U l f ( ~ , ~ ) ( 9 1 V ,gnV) ,... = f(glU,....g~lU). (2.14.6) In the special case in which V = 1 so G/V = G we have finf(u,l)given by

We shall now apply this to Galois groups of possibly infinite Galois extension fields. Thus suppose FIF is algebraic, separable and normal over F. We shall be interested mainly in the case in which is the separable algebraic closure F, of F, that is. the subfield of separable elements of the algebraic closure p of F . Let G = Gal F/F with its usual topology (BA 11, sec. 8.6). Let E/F be a finite dimensional Galois subfield of F/F and let V = Gal FIE. Then V is a closed normal subgroup of G which is the kernel of the restriction homomorphism a --i a 1 E . This is surjective so Gal E/F = G/V. Thus V has finite index and hence is open. Conversely, let V be any open normal subgroup of G. Then V is closed and G / V is discrete and compact. Hence G/V is finite and if E = Inv V then E/F is finite dimensional Galois with V = Gal FIE. The multiplicative group E x is a module for G/V and so we can define the cohomology groups Hn(G/V, E*).Now let K / F be a Galois suhfield of E/F and let U = Gal F / K so V c U. We have the inflation homomorphism Hn(G/U, K X )G ~ H " ( G / V E , x). Let C be the set of finite dimensional Galois subfields of FIF. We partially order C by inclusion. Since any two finite dimensional Galois subfields of F/F are contained in a finite dimensional Galois subfield of FIF, C is a directed set. It is clear that the set of groups H n (G/V, E*) together with the set of inflation maps between any two such groups determined by finite dimensional Galois subfield E and K with E > K satisfy the conditions that permit definlng the direct limit H:(G, F * ) = lim H7'.(G/V.E*) +

(Theorem 2.8 of BA 11). We call this group the n-th C O T L ~ % T L U ~cohomology US grou~pof G with coeficients in F".We can also give a "global" definition of this

2.14. Isomorphism of Br(F) and H ' ( F )

93

group. For this purpose we note that F* is a continuous module for G in the sense that for fixed a E F*the map a --i aa of G into F*is continuous relative to the topology of G and the discrete topology of F*. Now let c ~ ( G ,F * ) be the group of continuous maps of the n-fold product G x G x . . . x G into F * . The coboundary operator maps C; (G, F * ) into C?+'(G, F) so we can define the corresponding cohomology groups. It is not difficult to show that these groups are Isomorphic to the continuous cohomology groups defined as direct limits We refer the reader to Serre's monograph Col~omolog~e Galozsz~nne ([64]) for a more complete discussion of continuous cohomology of profinite groups (= inverse limits of finite groups). The groups G are instances of such groups. For our purposes it will be convenient to use the definition by direct limits. 1% shall now show that H:(G, F*) B ~ ( F / F ) . We recall that if E/F is finite dimensional Galois the map [k] --i [ ( E ,G, k ) ] is an isomorphism of H 2 ( G .E * ) onto B r ( E / F ) (Theorem 2.3.18) (iii)). If K / F is a Galois subfield of E/F and V = Gal FIE and U = Gal F/K then Theorem 2.13.8 implies the commutativity of the diagram

"

H2(G/U. K * )

+

Br(K/F)

H2(G/V,E*)

+

Br(E/F)

where the horizontal maps are te isomorphisms we have noted. Since every finite dimensional central simple algebra split by F is split by a finite dimensional Galois subfield of E', B ~ ( F / F )= U E E CB r ( E / F ) . Thus B ~ ( F / F )can be regarded as a direct limit of the B r ( E / F ) . It follows readily from the commutativity of (2.14.8) and the definition of direct limits (BA 11, p. 70) that WP have Theorem 2.14.9. H:(G,F")

" B~(F/F).

The important special case of the foregoing theorem is that in which = F,, the separable algebraic closure of F. In this case we abbreviate HP(G, F,') to H n ( F ) . Moreover. since every finite dimensional central simple algebra has a separable splitting field, Br(F,/F) = Br(F). Hence we have Corollary 2.14.10 H 2 ( F ) r Br(F).

Now let e be a positive integer not divisible by the characteristic of F. Let Br,(F) be the e-torsion part of Br(F), that is, the subgroup of classes [A] such that [Ale = 1. Let pe denote the subgroup of the multiplicative group of F" of p-th roots of 1. It is clear that p, c F,. By 2.13.14. if [A] E Br,(F) then [A] = [(E,G, k ) ] where the k,,, E p e . Let C , be the set of finite dimensional Galois subfields of F, that contain p,. If E/F E C, and V = Gal F,/E then Gal E/F G/V where G = Gal F,/F and we have an induced action

"

94

11. Bralier Factor Scts and Noether Factor Sets

on pr of the Galois action. Hence we can define H2(G/V. p,) by this action. The iwmorphism of H'(G/V, p,) onto Br,(E/F) = B r ( E / F ) n Br,(F). As in (2.14.8). we have the commutative diagram H2(G/li. k ~ e )

+

Br,(K/F)

if K: E E C, and K c E. If we define Hz(G; ,&H2(G/V,p,) then the commutativity of (2.14.9) implies, as in the proof of Theorem 2.14.9 the following Theorem 2.14.12. H:(G, p,)

"Bre(F)

If p, C F thrn the action of G on p, is the trivial one and H:(G, p,) is the usual contirluous cohonlology group of G with coefficients in p,.

111. Galois Descent and Generic Splitting Fields

Let A be a central simple algebra over F split by a finite dimensional Galois ~ = EndEV where V extension field E/F with Galois group G. Then E $ 3 A is a vector space over E of dimensionality the degree m of A I F . If 0 E G, o determines the automorphism a , of EndEV that is the identity on A and is c on E . The a , form a group arid it is clear that A = Inv a the set of fixed points of the a,. Thus A can be obtained by "Galois descent" from the split central simple algebra EndEV. Now a , has the form 1 --i u,!uil where TL, is a oscnlilinear transformation of V and since u, is determined up to a multiplier in E* we have u,u, = k,~,u,, for k,,, E E*. Then the k,~, constitute a factor set k from G to E x . The I L , can be used to define a transcendental extension field F,(k) of F in t,he following way. Let E ([) = E([1, . . . ),[ wllere the <,are indeterminates and identify the E-subspace CE<, of E ( < ) with 'I/ = C E z , , (zl.. . . :z,) a base for VIE. Then the u, can be regarded as semilinear transformations of C E [ , and u, has a unique extension to an ilutornorphisni ~ ( a of ) E ( < ) / F sucll that ~ ( a I) E = 0 . This restricts to ) ~the subfield E([)o of rational functions that are an automorpllism ~ ( c ' of llomogeneous of degree 0 in the sense that they are quotients of homogeneous ( Tor)^ ) ~ (but polynomials in the ['s of the same degree. TVe have ~ ( O ) ~ V = . we have the subficld F,,( k ) = Inv q(G)o which not 77(o)q(~)= ~ ( o r ) )Hence we call a Brauer field of the central simple algebra A. The field F,(k) is a generic splitting field for A in a sense defined in Section 3.8. Such fields were first studied for quaternion algebras by Witt ([34]) and for arbitrary central simple algebras by Amitsur ([55] and [56]).Further results and a simplification of the theory are due to Roquette ([63] and [64]).

.

One obtains a deeper understanding of the relation between A (or the Brauer class [A]) and F,(k) by introducing a geometric concept, the BrauerSeveri variety of A that was first considered by Ch6telet ([44])in characteristic zero prior to Amitsur's paper. Our approach will be the geometric one and we shall develop the rudiments of algebraic geometry that we require. This will include a discussion of the Pliicker equations for Grassmanniarl varieties. After introducing Galois descent for algebras in Section 3.1 and for fields in Section 3.2 we formulate the result,s in cohomological terms in Section 3.3 and we define the basic "long" exact cohomology sequences for H i ( G , A), 0 5 i 5 2, where G is a group and A is a set on which G acts. In Section 3.4 we define Grassmannians and determine defining equations for these varieties. N. Jacobson, Finite-Dimensional Division Algebras over Fields , © Springer-Verlag Berlin Heidelberg 1996, Corrected 2nd printing 2010

96

111. Galois Descent and Generic Splitting Fields

These are used in Section 3.5 to define the Brauer-Severi variety V A of a central siniple algebra A. In Section 3.7 we show that the field F',(k) defined above is isomorphic to the field of rational functions F(VA)o on V A . In Section 3.8 we define generic splitt,ing fields and show that F ( V A ) iis such a field. We then switch back to Brauer fields arld use these to derive the principal results of Amitstir and Roquette on generic splitting fields (Sections 3.9: 3.10). In Section 3.11 we tiefine anot,her georrletric concept,, the norm hypersurface SA of a central simple algebra. This provides another generic splitting field for A. The results here are due to Heuser ([78]) and to Saltman (1801). Still another projective variety, the variety RA of rank one elements of A p , F an algebraically closed extension of F, is considered in Section 3.12. These have been studied by Petersson ([84]) and by Jacobson ([85]). They provide st,ill another generic splitting field for A. The relations connecting the varieties V A ,SA and RA and the corresponding generic splitting fields (due to Petersson) are given in this section. In Section 3.13 we define the important concept of the corestriction of algebras using a method based on Galois descent.

3.1. Galois Descent for Vector Spaces Let V be an m-dimensional vector space over a field E. F a subfield of E. We drfine an F-form of V to be an F-subspace Vo such that the canonical map

Ca, R x,

--i

C a i x i ,a, E E, xi E Vo

(3.1.1)

is an isomorphism of E 8~Vo onto V. This is equivalent to: any base for Vo/F in a base for VIE. Hence we obtain all the F-forms by choosing bases ( 2 1 . . . . z,) for V I E and taking Vo = Cy Fx,. Now suppose E is finite dimensional Galois over F with G = Gal E I F . Let Vo be an F-form of V with base (XI... . , x,) over F and let a E G. Then a defines a map

.

m

m

of V into V. This is bijective and a-semilinear and we have

Thus ol = { a , I a E G) is a group of semilinear transformations in V such t>hata, is a-semilinear, a E G. Moreover, we can recover Vo from a by the formula Vi = {y E V / a,y = y, a E G). (3.1.4) We have seen in Lemma 2.13.1 that, conversely, if a = { a , / a E G) satisfies (3.1.3) and a, is a-semilinear for E G then Vo defined by (3.1.4) is an F-form. It is clear also that if ( X I . . . ,x,) is a base for Vo/F then the

glven a , is the map defined by (3.1.2). Our results establish a 1 - 1 correspondence between the set of F-forms of V and the set of groups of senlilinear traiisformations a = {a,) satisfying the stated condition. Let Vl be a second form of V and let 0 = (8,)be the corresponding group of serriilinear transformations. Suppose X is an F-isomorphism of Vo/F onto Vl/F. This has a unique extension to an automorphism X of VIE. For any Vl = I",. It a. Xo,Xpl is a g-semilinear trarlsformatjon such that Xa,X-l follows that P, = X c u , X - l , a E G. In particular. if Vl = Vo then we see that the group G!(Vo) of bijective linear transformations of V,/F is isomorphic to the subgroup of Ge(V) of bijective linear transformations that commute with every a,, a E G. All of this is rather trivial. Sornethirig more interesting arises if V has some additional structure that we wish to carry over to Vo. Before formulating this in a general way we consider a couple of examples.

Algebras Let A be an a,lgebraover E (not necessarily associative). We define a semil.inenr autom,orphisrn of A / E to be an automorphism of A as algebra over F that of A as vector spa,ce over E. It is clear that is a senlilinear tran~format~ion the automorphism in E associated with a semilinear automorphisnl of A / E is contained in G = Gal E/F. Let A be the group of seinilinear automorphisms of A I E , A the subgroup of automorphisms of A/E. Then Aa A. We define an F-form of the algebra A / E to be an F-form of the vector space A that is a,n F-subalgebra of A. Such a forrn exists if and only if there exists a subgroup a = {a, / a E G ) of A such that a, is a-semilinear. Then A, = { a E A / a,a = a , a E G ) is an F-form of the algebra A and all F-forms of A are obtained in this way. The group a = {a,) is called a section of A relative to A. It is clear that A is a semidirect product A M a of A and a :AaA. ; i = A a = a A , A n a = l . We have a 1 - I correspondence a --, A, between the set of sections of relative to A and the set of F-forms of A. Moreover, if a and P are two sections Ap as algebras over F if and only if a and /3 are A-equivalent in then A, the sense that there exists a X E A such that = XaXP1.Hence we have a 1- 1 correspondence between the isolnorphisrn classes of F-forms of A I E and the equivalence classes of sections of A relative to A. Finally, we note that the group of automorphisms of a form A,/F is isomorphic to the subgroup of A of the X E A such that XaX-l = a . We now specialize to the case A = lZfm(E). Then if A, is an F-form of nifm(E). This implies that A, is central the algebra iLfm(E), E 8~ A, simple over F of degree m split by E . We have A, = bfT(D), D a central division algebra of degree d. Then m = r d . Thus the index of A, is a divisor of m. On the other hand, if B is a central simple algebra over F of degree m split by E then E E F B = lZfm(E)and B is isomorphic to the F-form l@B=(lz~blb~B}ofM~(E).

"

"

98

111. Ga,lois Descent and Generic Splitting Fields

The autoniorphisnis of A&(E) are inner, that is, they have the form X --i UXUpl where U E GL,,(E). Also, if a E G then X --i a X is an autoinorphism of A/l,(E) as algebra over F and is a-semilinear. It follo~vsthat the seinilinear a.ut,omorphisms of !I/l,(E) liave the form X --i U(aX)UP1. Thus the set of these is the group of semilinear autornorphisrns of hJ,,(E) and the set of inner automorpllisms is the normal subgroup A of A. lye now switch from A/l,(E) to the isomorphic algebra EndEV where V is an m-dimensional vector space over E. Then A is the set of autornorphisrns L --i II,E?L-~ where u is a 0-semilinear trarlsforrr~atioriof V for some a E G and il is the normal subgroup of those for which u E EndEV. Let a = { a , 1 a E G) be a section of (1 relative to A. Then a , : L --i u,Lu;l where u , is a asemilinear transformation of V and u, is determined up to a multiplier in E x . The condition a,a, = a,, gives

Eu, is ari algebra of liriear transforrnatiorls of V / F and the set B, = CaEG determined by a that is isomorphic to the crossed product ( E , G, k). The Fform of EndEV determined by a is A, = { e E EndEV / a,! = !.a E G). As me h d ~ enoted before. this is just the centralizer of B, in EndFV. It follows that A, B: and [B,] = [A,Ip1. Hence the index of B, is also a divisor of m. N

Polynomial Functions on a Vector Space Let V he a vector space over an infinite field E and let ( X I ,. . . . x,) be a base E, indeterminates. Then f defines for VIE. Let f (El. . . . ,Em) E E[[l.. . . .<,I. the polynomial map a = Ca,x, --i f ( a l , .. . .a,,) on V. Since E is infinite, distinct polynomials in the ['s determine distinct maps. Let a be a group of semilliiear transformations a = { a , I a E G) such that a , is u-semilinear and let V, be the corresponding form of V. Then f (V,) C F if and only if In this case f / V, is a poly~lornialfunction on V, defined by an element f, E F [ t l . . . . .<,I and a hdse (21,. . . x,,) for V,/F. We shall call (V,. fa) forrrl o j (V, f ) M'e have a 1 - 1 correspondence between the forms (V,, f,) of (V. , f ) and the a = {a,) satisfying (3.1.6). We define an zsomorphzsm X of the form (V,, fa) onto the forrn (VJ, fo) to be a bijective linear transformation X of Va/F onto VD/F such that

In this case the extension of X to a linear transformation of VIE satisfies f (Xa) = f (a), a E V and p, = Xa,X-' for the groups P = {p,) and a = {a,). Suppose that forms of (V, f ) exist. The simplest way of assuring this is to begin with a polynomial function g on a form W of V. Then if we define ,f to

3.2. Forms of Fields

99

be the extension of ,9 to a polynomial function on V: it is clear that (IV, g) is a form of (V, f ) . Let A be the group of bijective a-semilinear transformations ( 2 , of V I E s ~ i c h that f (a,n) = a f ( u ) ,a E V for all a E G a,nd A the normal subgroup of linear transformations contained in A. Then A a A and is the serliidirect product of A and any ct corrcsponding to a form of (L7' , f ) . An inlportant special case of polynomial maps is that in which f = Q is a homogeneo~lspolyno~nialof degree two. The corresponding function Q is a quadratic form and A is the orthogonal group O(V, Q). An interesting special case of polynomial functions tha,t ha,s been studied only recently (Waterhouse 1841 and Jacobson [84]) is the following. Let V = M m ( E ) and let f be the determinant function: If X = (Eij), Ez3 indeter~ninates. then det X is defined as usual and the corrcsponding map is a = (at3)--i det a. Let A he a central simple algebra over F of degree m split by E and let n be the reduced norm on A (see sec. 1. G ) . Then (A. n) is a form of (Adm, (E):det). Not all F-forms of (lZfm(E):det) are obtained in this way. For example, let F = R,E = C , and let H be the set of m, x r n hermitian matrices with entries in C . Then iZJm( C ) = C H C ~ ' C R H and (H, det) is a real form of (lZfm( C ): det). The two examples we have given are special cases of the following situation. Let V be a finite dimensional vector space over the finit,e dimensional Galois extension EIF, G = Gal E I F . Let A be a group of semilinear transformations of V such that if A is the normal subgroup of linear transfornlations contained in A then A is a semidirect product A x a of A and a subgroup a = {a, / a E G) such that tr, is a-semilinear. Then a1 = 1 and a,a, = a,,. We shall call a a section of A relative to A. Associated with a: we have the form V, = {x E V I a,z = 2: a E G) of V. We shall call this a A-form of V. If V, and V8 are two A-forms of V determined by the sections a and ,!il respectively then a bijective F-linear map X of V, onto Vpis called a A-isom,ol-ph,ismif the (unique) extension of X to a linear transformation X of V is contained in A. Then 13, = X a , X p l , a E G. In this case we say that ,8 and a are A-equivalent. We have a 1 - I correspo~idencebetween the A-isomorphism classes of Aforms and the A-equivalence classes of sections of A relative to A. Finally. we note that the group of A-automorphisms of a A-form V, is isomorphic to the = a,. a E G. subgroup of A of the X's such that Xa,X-'

"

3.2. Forms of Fields We consider next a counterpart for fields of the theory of forms of a vector space that was presented in the last section. As before, let E/F he finite dimensional Galois, G = Gal E / F and let K be an extension field of E. not necessarily finite dimensional over E. In the applications K will be tranicendental over E. Since E/F is Galois any T E Gal K / F stabilizes E and hence Gal K / E a Gal K / F .

(3.2.1)

111. Galois Descent and Generic Splitting Fields

100

Now let I? be a subgroup of Gal K I F . A = A n Gal K / E so A a A. By a sectzon of 2 relatzve to A we shall mean a subgroup q = { ~ ( a )a E G) of A such that q(a) / E = a , a E G. It follows that

that is,

/i=Axq.

(3.2.3)

Now let K, = I11v q. Then K/K, is Galois with Gal K/K, = rj so [ K : K,] = n = q = jG/. Let a E K, n E and a E G. Then a = ~ ( a / )E and q(a)a = a since a E K,. Thus aa = a for all a E G and hence K, n E = F. Then K , E = K, E F E in the sense that the cariollical map of K, 8~E into K,E is an isomorphism (Section 2.13). Then [K,E : K,] = [E : F] = IGI and hence K = K,E. Thus (3.2.4) K = K, ZF E and

q = Gal KIK,

c I?.

(3.2.5)

We shall now fix and we shall define a A-form of K(A = I?n Gal K / E ) to be a subfield K o / F of K / F such that K = KOE = KO@ F E and Gal K/Ko C I?. We have shown tha,t if q is a section of A relative to A then K, = Inv q is a il-form. Conversely, let I(o be a A-form of K . Then any a E G has a unique extension to a,n autornorphism q ( a ) = I 8 a of K/Ko. Then 7 = {rj(a) / a E G) is a section of I? relative to A and Inv a = KO. Thus we have a 1 - 1 vorrespondence between the set of A-forms of K and the set of sections of A with respect t,o A. If K, and K C are A-forms corresponding to the sections q and respectively then any isomorphism of K , / F onto K C / F has a unique exterlsioli to an automorphism X of K I E . We shall call the given isomorphism of I(, onto K c a A-isomorplzism if X E A. If there exists such an isomorphism t,hen K, and K C are A-eq~~ivalent. This is the case if and only if /3 = X a X - ' for some X E il.It is clear also that the group of A-automorphisms of K, is isomorphic t,o t,he subgroup of A of the X such that XqX-I = rj. Sincc A n q = 1 this is equivalent to Xrj(a) = q(a)X, a E G. We now consider the fields that are of interest for the study of Brauer groups. As before, let E / F be finite dimensional Galois with G = Gal E / F . Let E ( < ) = E(E1, . . . .Eln) be the field of rational functions in V L indeterminates over E. Then E([) contains the polynomial ring E[[] = EIS1,. . . ,Ern] which has the usual grading by (total) degree in the 6's. If f and g are homogeneous polynomials of degree r and s , respectively, then fg-' is a homogerleous rational function of de,gree r - s. It is clear that the set of homogeneous rational funct,ions of degree 0 constitutes a subfield E(<)o of E ( E ) I E . If f E E(<)o: f (El,. . . ,Crn) = f ( E I E ~ '.~. ,Ern-lE,', . 1). It follows that E([)o = E([i,. . . , and the are algebraically inde6; = pendent over E . Let F(E) = F(E1,. . . ,Ern) and F([)o the field of homogeneous rational functions of degree 0 in F(<). Then F ( o o = F ( [ i , . . . ,[A_,), E; = ti[;' and is the field of fractions of F[[i,.. . , EL-,]. Now F([i,. . . ,

<

I[

3.2. Forms of Fields

101

Then E F ( < ) o is a domain and since this is finite dimensional over the field F([)o it is a subfield of E(<)o. Hence EF([)o contains the field of fractions of Since this field of fractions is E([)o it follours that El<:. . . . ,

Wc now take E([)o for our extension field K of E. Let V = C y EE,, the subspace of homogeneous elements of degree 1 of E[<]and let u be a bijective a-sernilinear transformation of V for u E G. Then u has a unique extension to an automorphism of E ( ( ) / F .This has the form

where the a on the right indicates the application of a to the coefficients of f ( t l . . . . .Em). It is clear that (3.2.8) stabilizes K = E(E)o and its restriction to E is a . The set of automorphisms of K obtained in this way constitute a subgroup of Gal K / F which we shall take as our group A. Since the restriction of (3.2.8) to E is a it is clear that A is the subgroup of -4 of the automorphisms determined by the u E EndEV. Now let u,, a E G. be a bijective a-semilinear transformation of V and suppose (3.2.9) ~ , ' t 1= ~ k,,,'t~,,, 0 , T E G. k,,, E Ex. Let 77(u,) denote the automorphism of E(E) defined by (3.2.8) and let T ( I L , ) ~= ~ ( u , )I K . Then q(u,)o I E = a and

Hence T ~ ( G = ) ~{ ~ ( u , ) I~ a E G) is a section of A relative to A and I = Inv v ( G ) ~is a A-form of K. Such a form will be called a B r a u p r field of dzmenszon m - 1. The general theory of A-forms of fields is applicable to Brauer fields. 117~recall that an extension field K / F is called separable if it is either of characteristic 0 or is of prime characteristic p and it satisfies one of the following ecluivalent conditions: (1) Every finitely generated subfield K ' I F is separably generated. that is, K 1 / F has a transcendency base B' such that K' is separable algebraic over F ( B 1 ) . -1 (2) K and FP-' are linearly disjoint over F (that is, K F P - ~ K @ F FP in ~p-l). (3) K and F P - ~ are linearly disjoint over F (Section 8.15. BAII).

"

If K / F is separable then any subfield K 1 / F is separable. If K I F is finitely generated then K / F is separable if and only if it is separably generated. As extension field K / F is called regular (in the sense of Weil) if it is separable and F is algebraically closed in K .

102

111. Galois Descent and Generic Splitting Fields

Theorem 3.2.11. Any Brauer field 2s regular over F . Proof. Since E ( O o = EKv, = E ( < i : . . ,ELp1)= EF(<$.. . . :ELp1),[,'= (by 3.2.7), E([)"= F ( [ ; ,. . . . ELp1) (E) is finitely generated and separably generatcd over F. Hence E([)" is separable over F and so it is subfield Kv, . Clearly E is algebraically closed in E ( [ i, . . . , [A_,). Hence the elenleiits of K,, that are algebraic over F are contained in E i?K,, = F. Thus F is is regular over F . algebraically closed in K,, and

[,E;'

The importance of the property of regularity is that if K j F is regular and F1 is any extension field of F then F1 % F I< is a domain (see e.g. BA 11. Theorem 8.51, p. 550). In general, if F1@lFK is a donlain then we denot,e the by F1.K.XTeremark that F'.K = F'K = F ' 8 F K field of fractions of F ' E ~ K if either F' or I< is finite dimensional over F. If K is regular then F' . K is a free cornposite of F' and K in the sense that if C and D are algebraically independent subsets (over F ) of F' and K respectively then C n D = 8 and C U D is algebraically independent. Up to isomorphism F' . K is the olily free composite of F' and K (see BAII. Sec. 8.19: p. 550f).

3.3. Forms and Non-commutative Cohomology As in Section 3.1. let V be a vector space over the finite dirrlerlsional Galois extension E I F , G = Gal E j F and let be a group of semilinear automorpliisms of V such that the subgroup A of linear transformations contained in A has a section a. We now fix a section a" and use this to define an action of G on il by OX = a o U X a ~ ~ . (3.3.1) We have ( ~ 7 r ) X = a ( r X )1.X = X

(3.3.2)

a ( X l X 2 ) = (~7X1)(~7X2)

(3.3.3)

and for A, A, E A, a. T E G. Thus the action is by automorphisms of the group A. Next, let a = { a , / a E G ) be any section of with respect to A. Put

-1 1 1 Then h, E A and h,, = a,,aOu7 = a , a , a ~ a & = a , a ~ ,ao, (a,az)a&! = h, (oh,).Thus we have

Hence h : a --i h, is a crossed homomorphism of G into A. On the other hand. we can re-trace the steps and show that if h is a crossed homomorphism of G into A then n = {a,) where a, = h,ao, is a section of A with respect to A. The correspondence we had between A-forms of V and sections now

3.3. Forms and Non-commutative Cohomology

103

yields a bijection of the set z l ( G , A) of crossed homomorphisms of G into A with the set of A-forms. Since a, = h,,ao, it is clear that the A-form corresponding to t,he crossed homomorphism 11 is the F-subspace of V of x's such t,hat l ~ , a ~ , n = : 2:: a E G. If Lrl and IT2 are two A-forms of V corresponding to the crossed holnomorphisms hl a,nd hn respectively then we have seen that Vl and V2 are A-isoinorphic if and only if there exists a X E A such that ha,ao, = X(hl,ao,)X-l. This is equivalent to hn, = X ~ ~ , Q ~ , X - ~ = L YXhl, ~ ~ (oXjpl. Thus Vl and V2 are A-isomorphic if and only if there exists a X E A such that

This relation on crossed homomorphisms is an equivalence relation. The set of equivalence classes defined by the relation (which need not be a group) is denoted as H1(G. A). Our results show that we have a 1 - I correspondence between H1(G, A) and the set of A-isomorphism classes of A-forms of V. It is clear also that the group of 11-automorphisms of the A-form Vl corresponding to h E Z1 (G. -4) is isomorphic to the subgroup of A of the X's such that

The nlap X --, X / Vl is an isomorphism. IVe can make a similar shift from sectiolis to crossed homomorphisms in dealing with A-forms of a field as in Section 3.2. Here we have a field K / F containing a subfield EIF that is finite dimensional Galois with Galois group G and we have a subgroup A of Gal K I F such that if A = A fl Gal K I E then there exists a section of A re1at)ive to A. As in the vector space case me pick a particular section qo and use it to define an action of G on A by OX = r/o(o)~770(a)p1. Then if q is any sect'ion of A relative to A, 7 defines a crossed homomorphism h = {h, / a E G) where h, = rl(a)170(a)-i. We have a 1 - 1 correspondence between these crossed homomorphisms and A-forms of K and a 1 - 1 correspondence between H1(G,A) and the 'A-isomorphism classes of A-forms of K . Finally, we have an isomorphism of the group of Aautomorphisms of the A-form correspondirlg to the crossed homomorphism h with the subgroup of X E A such that (3.3.7) holds. We shall now formulate these ideas in a more abstract forin and apply a fragment of colroinology theory to obtain an exact sequence that is useful for the study of the Brauer group. If G is a group, a group A (not necessarily abelian) is called a G-group if G acts on A by automorphisms: ( g l g z ) ~= yl(gzx), 13: = 2, g(zixa) = (gzl)(gzz) for g,g, E G, z , x i E A A lzomomorphzsm (or G-homomorphzsm) 7 of a G-group A into a G-group A' is a group homomorphism such that ~ ( g x = ) g(qx), g E G, x E A. If A is a G-group we define the 0-th cohomology group ofG wzth coeficzents in A by

104

111. Galois Descent and Generic Splitting Fields

As in the special cases above, we let Z1(G, A) denote the set of crossed homomorphisms of G into A and we define equivalerlce for these as in the special Lases: if u, E Z 1 ( G , A ) . z = 1,2. then u2 is equzvulent or cohomologous to ul if there exists an a E A such that

u2(g) = aul(g)(ga?-', g E G.

(3.3.9)

The set of equivalence classes in Z1(G: A) relative to this relation is denoted as H 1 ( G , A) and is called the first cohomology set of G with coeficien,ts in A. H1(G, A) is riot a group if A is not abelian. However, it does have a hit of structure given by a distinguished element. In the set Z1(G, A) we have the eicment 1 such that l ( g ) = l Athe unit of A for all g E G. This deterniines the equivalence class [I] in H 1 ( G , A ) . Evidently this is the set of crossed homor~iorphisnisu such t,hat u,(g) = ~ ( ~ a ) for - l some a E A. (In the special case we considered above, [l]corresponds to the forms that are A-isomorphic to the A-form Vo.) We shall not define higher cohomology sets H n ( G . A) for n > 1 unless A is abelian, in which case the usual definitions apply. Now let q : A + A' be a homomorphism of G-groups. Then gx = x implies g(qz) = qx so we have the homomorphism v0 : H O ( G , A ) + H O ( G ,A') obtained by restricting the domain and codomain of 7.Now let u E Z1(G. A). Then applying q to the defining corlditions u(gh,) = u(g)(gu(h)) gives qu(gh) = (qu(g))(gqu(h)). Hence r l l e ~defined by (qlu) (g) = q(u(g)) is an elernelit of Z1(G. A'). If a E A then 17(au(g)( g a ) ' ) = (qa)((qu)(9))(g(qa))-I so equivalent elements of Z1(G. A) are mapped into equivalent elements of Z1 (G; A'). Hence we have the induced nlap q1 : H1(G, A) + H1(G, A'). It is clear that this maps the distinguished element [I] of H1(G, A) into [I] of H1(G, A'). Now suppose we have an exact sequence of G groups

Then it is easily seen that the sequences

are exact where exactness in the first instance is defined as usual and in the 5econd instance it again means that the inverse image of [l]under p1 is the image of H1(G, A') under i l . In general, in dealing with a sequence of maps of collomology sets that map the element 1 of any set into the element 1 of the next then we define exactness at one of the sets to rnean that the image of the preceding set coincidcs with the inverse image of 1 of the following set. This coincides with the usual definition in the case of groups. Now let a" E HO(G.A") so a" E A" and gal1 = al'.g E G. Since p is surjective we have an a E A such that pa = a". The element a(ga)-l satisfies p ( a ( g a ) l ) = (pa) (p(ga)-la"(ga")l = 1. Hence there exists a unique element u(g) E A'. It follows that u' : g * ul(g) is in Z1(G, A'). iVIoreover, a second

105

3.3. Forms and Non-commutative Cohomology

choice of n such that pa = a" must have the forrn (7b')a mhere b' E A'. The correspoliding element of Z 1 ( G ,A') is equivalent to u'. We therefore have a map AO of HO(G.A") into H 1 ( G .A') such that

mhere [u']is the equivalence class of u'. This maps 1 into [ I ] and one can verify directly that A' provides the connecting link to make

exact. UJe dssurne next that zA' = ker p is in the center of A. Then H2(G.A') is defined slnce A' 2 zA' is abelian. Now let u" E Z1(G,A"). Since p is surjective. ). for every g E G , we have an element u(g) E A such that p ~ ~ (=g ?) ~ " ( gThen

Hence there exists a unique element k l ( g ,h ) E A' such that

Direct verification, using the fact that

is in the center of A shows that ikl(g,h)kl(gh,!) = igkl(h,,!)kl(g,h!). Hence

Thus k' : ( g . h ) --t k l ( g , h ) E Z2(G.A'). If we replace u(g) by icl(g)u(g), v l ( g )E A' we obtain a 2-cocycle cohomologous to k'. The same thing happens if u" is replaced by an equivalent element of Z1(G.A"). It follows that we have a nlap A1 of H 1 (G,A") into H 2( G ,A') such that

-

[u"]

[k'].

(3.3.13)

This maps the class [ I ] into [ I ] and it is straightforward to verify that

is exact. This is the analogue of the usual long exact cohomology sequence. We now specialize to the case that arises in sorting out the forms of l14m(E) or, equivalently. the central simple algebras of degree r n split by E . Since the autornorphisms of M m ( E ) are inner we have the exact sequence 1

+ E*

--t

G L m ( E )+ Aut M m ( E )--t 1.

(3.3.15)

111. Galois Descent and Generic Splitting Fields

106

Now we have the F-form i\I,,(F) of AI,(E) and me use the coiresponding section a in Aut AI,(E)/F relative to Aut hIn,(E) to define the action of G on Aut 1I,, (E). Using this and the usual action of G on E* and GL,,(E). (3 3.15) becomes dri exact sequence of G-groups. The corresponding exact sequence (3.3.14) is 1

-

F"

i

GL,(F)

+ H I ( G , GL,(E))

-

+

Aut hl,,(F)

no

+ H'(G,

H ~ ( G~. u nr,(E)) t

E*)

S H?(G. E-i

(3.3.16)

By Speiser's theorem (Lemrrla 2.13.4). H1(G, E*)= 1 and H1(G. GL,(E))

=

I. Hence we have the exact sequence

If Aut 3I,,(E) were abeliarl then the exactness of (3.3.17) would imply that A' 1s injective. However. since Aut h I m ( E ) is not abelian we can not conclude that A1 is injective in this way. We could conclude the result using the correspoiidence between H'(G. Aut A17,,(E)) and forms. However, we shall gixe an ~ndependentproof. JVe note first that, by (3.3.15), we can identify Aut llIn,(E) with the projecti\e linear group PGL,(E) = GL,(E)/EA. 4 1-cocycle from G to PGL,(E) is a map a --i p, of G into PGL,,(E) such that pOT= k ~ o ( o p TIf) .Uo E GL,(E) is a representative of ~ c , then we have

-

It is readily seen from the definitions that A1 maps the class of p : u --i Ir, in k,,, in Z2(G,E*). H1(G. P G L , ( E ) ) into the class of [k] where k is (a, T ) Sloreover. if o --, ,L;" a,nd a --, pi2) are two 1-cocycles of G into PGL,,(E) and c-:' is a representative of p c ) t,hen these cocycles are cohomologous if there exists ail L t GL,(E) such that

It follonrs that the injectivity of A' in (3.3.17) is a consequence of the following rnatrix theorem, which is an extension due to Scllur ([19])of Speiser's theorem. Theorem 3.3.20. Let EIF be ,finite dimensional Galois with G = Gal E I F . Suppose for every (T E G we have matrices I,':) E GL,(E), 1; = 1, 2 such that U?)(al~!~))= k,.,Ut,) where k,,, E E*. Then there exists an L E GL,(E) such that = LU;')(~L)-', a E G.

UP)

Proof. Let V be an m dimensional vector space over E and let ug) be a 0semilinear transformation of V having the matrix relative to a chosen base ) ( e l , . . . . e,,) of VIE. Then up)a = (aa)u$) and u ~ ) u=~ k), . , ~(,2 ~ , a , E~G. Now let ( E . G , k) be the crossed product CEu, where u,a = (aa)u,. u,u, = k,,,u,,. Then (E.G, k) is simple and we have two representations

UP)

3.4. Grassnlannians

107

of ( E . G. k ) acting in V I F . These are equivalent. Accordingly, we have a bijective linear transformation t of L7/Fsuch that

In particular. we have tat-' = a or a t = Ea for a E E. so 1 is a linear transformatioli of VIE. Then if L is the matrix of E relative to ( e l , . . . . e m ) the relation (3.3.19) is a consequerlce of the facts that uL2) = !trL1)tpl and I/':) and L are thp matrices of u:) and 1 relative to (el.. . . e m ) .

.

It remains to describe the image of A' in H 2 ( G ,E * ) .If A is the F-form of ilir,(E) determined by an element of Z1(G,Aut lLf,(E)) then we have see11 that the index d of A is the same as that of ( E , G, k). Hence d / In. Conv~rsely,suppose d / m. Then m = r d and n = [E : F] = sd. Now the irreducible module for (E.G, k) has dimensionality sd2 over F and we have a module V for ( E , G, k) of dimensiorlality r s d 2 over F. Then V is m n dimensiorlal over F and hence [V : E] = m. If me identify (E,G, k) wit,h the corresponding elldomorphisms of V and we choose a base for V I E me obtain rrlatrices U, E GL,(E), a E G, such that Uu(aUT)= k,.,LTU:,,. Thus o --, U, determines an element of Z1(G,Aut !Wm(E))and an element of H 1 (G. Aut Mrn( E ) ) whose image in H z (G, E*)is [k]. We shall now define the index of an element [k] of H2(G.E x )to be the index of the corresponding elerrlent [(E,G, I;)] of B r ( E / F ) . Then we have

Theorem 3.3.21. The m a p A1 of H1(G, Aut h f m ( E ) ) (H1(G, PGL,(E)) into H 2 ( G .E*) is injective and its image is the set of [k] ~ u h ~ o szndices e are divisors of m .

3.4. Grassmannians In the next section we shall establish an important connection, first noted by ,4. Chiitelet ([34]), between central simple algebras and certain varieties in projective spaces now called Brauer-Severi varieties. With any central simple algebra A of degree m we car1 associate a Brauer-Severi variety V A defined to be a certain subvariety of the Grassmannian defined by the m-dimensional subspaces of AF, F an algebraically closed extension of F. As a preliminary we need to recall the definition of Grassmannians and determine their defining equations. Our account will be self-contained and concrete. Further information on Grassmannians can be found in Hodge and Pedoe's [47]: Vol. I , Chap. VII. Let V be an n dimensional vector space over a field F and let A(V) be the exterior algebra of V with its usual grading: A(V) = BdAd(V) where A d ( v )

108

111. Galois Descent and Generic Splitting Fields

is the subspace spanned by the decomposable vectors wl A u12A. . . A wd, w, E V. AO(V) = F (see e.g. BAI, sec. 7.2). If W is a d-dimensional subspace of V with base (lol! u ; ~.,. . :wd) t,hen we call associate with W the non-zero ~ . . . A wd of ild(V). If ( I L I ~ ,. . . : w&)is a second decomposable vector w = U J A base for W then ui; A . . . A wh is a non-zero multiple of w. Hence we have a map W Fw of the set of d-dimensional subspaces of V into the set of one dimensional subspaces of Ad(V) spanned by the decomposable vectors # 0. Evidently this map is surjective. It is also inject,ive since we can recover W from w as the set of vectors n: such that

-

1%'~ define the Grassrnarlnzan Gr(n, d) (or Gr(V, d)) as the set of one dimensional subspaces F J of Ad(V) where w = wl A , . . A wd # 0. Let ( P ~e2. . . . . , en) be a base for V. Then we have the corresponding base

for Ad(V) consisting of the vectors

Thus [Ad(V) : F] =

(3

. If w is an element of Ad (v) then

w =

pil . . .id e,, A . . . A e,,

)

and the p,, . . .,, E F are coordinates for w called Plucker coordznates (relative to the given base ( e l , . . . . en) of V ) . In particular. we have Pliicker coordinates for an~7non-zero decomposable vector w = wl A. . .Awd which we can calculate explicitly. For. let n

and let a = (a,,) the d x n matrix expressing the w, in terms of the base ( e l . . . . , e n ) . For 1s ik 5 d, 1 5 J ( n put

AZJ...iT = 31"'3~

1

<

ail(il

:

az,31

...

...

1.

a2~j7.

! 'tT.IF

(3.4.4)

Then it is readily seen that the Pliicker coordinate pil . . .td for w = w1 A. . .Awd is p,L = A;;::;~. (3.4.5) It is convenient to extend the definition of the Pliicker coordinates of any w E Ad(V) by putting p,, . . . j , = 0 if any two of the indices are equal and otherwise defining

where {il , . . . , id) = { j l , . . . . jd) and il < . . .

< id.

3.4. Grassmannians

109

Now let

then

and

ix indicates the omission of the index i A .Hence rc A w

=0

if and only if

d for all 1 5 in < . . . < id 5 n . Hence, by (3.4.1) if W = El F w , is a ddimensional subspace then x E W if and only if (3.4.7) holds where the p's are the Pliicker coordinates of w = wl A . . . A wd. We shall use these equations to derive a system of equations on the Pliicker coordinates of w E A d ( v ) that are necessary and sufficient conditions for w to be decomposable. To derive such a system of equations we first determine a certain set of points on W = Eld F w , from the Pliicker coordinates of

w =w1

A , . .A W d .

For any i l < . . . < i d let ITi,...,,-, denote the ( n - d + 1)-dimensional coordinate subspace of V consisting of the vectors Cx,e, such that xi, = ... - x,,_,= 0. Then we have the

Proposition 3.4.8. The vector

Proof. It is clear that wi ,... i d - , E 1T,,... i ,-,. To show that w ,,..., ,_, E W we note that, n

d

-

C(-l)d+iAl...;;d 2:...Zd&1wi. z=1

Hence this vector is contained in W.

111. Galois Descent and Generic Splitting Fields

110

If we apply (3.4.7) to the vectors

<

IL~,,

<

,,_, we obtain

<

for all 1 i l < . . . < idpl n , 1 5 jO < . . . < jd n,.These relations are called the Pliicker eqr~ationsfor the Plucker coordinates of a n element of Ad(V'). We have shown that they hold for any w = wl A . . . A lud # 0.Hence we have the necessity part of the following Theorem 3.4.11. Let (p,, ...,,) be the Plucker coordznates (determined by a base of V ) of an, element w E Ad(\'). T h e n (3.4.10) are necessary and suSJicient conditions that w i s decomposable.

Proof (Faulkner). For the sufficiency we may assume w # 0, so some p,, ,, # d # 0. Now 3.4.8 suggests clloosillg d linearly irldepcndeilt elements of the form (3.4.9) arld showing that is a niultiple of their wedge product. We take

0.For simplicity of notation, l+eassume pl

We have seen tlmt if w has Pliicker coordinat,es (p,, ...,,) then x A w = 0 if and only if (3.4.7) holds. Hence if the Pluclter equations hold t,hen we have Moreover, since w: involves e j and none of the other u:i involve this base vector, it is clear that the wi are linearly independent and hence w' = w: A . . . A ru: # 0. We now claim that the conditions (3.4.13) imply t,llat w is a multiple of w' and hence is decomposable. To see this we choose a base ( w i , . . . : w&,. . . , wk) for V. Then we have the corresponding base , 1 5 il < . . . < id n for Ad(V) and hence 1

If q,..,

<

<

# 0 for some ( i l , . . . , i d ) # ( 1 . 2 , . . . , d) then there is a j, 1 j 5 d, which is missing in (21, . . . , i d ) . Then w AW; # 0 for this j, contrary to (3.4.13). Thus LJ = ql...dw; A . II w; is decomposable. Example. The simplest case, Plucker's equations , that will be of interest in the sequel is that in which n = 4 and d = 2. Here (3.4.10) reduces t o the single equation

3..5. Brauer-Severi Varieties

111

3.5. Brauer-Severi Varieties MTe recall that if V is ail n-dimensional vector space over F then the one dimensional subspaces F v , E V are the points of the prodective space P V of 71 - 1 dimensions. If (el.. . . : e n ) is a base for LT/Fand v = C a i e , # 0 then (a1.. . . a,) are called homogeneous coordinates for p = F c (relative to the given base). These are det,erinined up to multipliers in F " . I f f = f ( < I ,. . . . En) is a hoinogeneous polynomial in Pi<]= F [ & ,. . . Ei indeterminates: then n7e say that Fu is a zero of f : and write f (p) = 0, if f ( a l , . . . ,a,) = 0. Since f is homogeneous this is independent of the choice of the homogeneous coordinates ( a l , . . . . a,,) of p. Now let F be a n algebraically closed field containing F and let S be a set of lionlogeneous polynomials in F [ ( ] .Let V(S) = {p 6 PVF I f (p) = 0, f E S ) . Then V = V(S) is called a projective F-r~larietyand S is a defining set of polynomials for V. For any such set S it is clear that V(S) = V(F[<]S) and F[<]Sis the ideal generated by S . Given V we define the ideal Z(V) of V t o be the ideal in F[[]generated by the set T of hornogeneous polynomials f E FIE] such that f (p) = 0 for all p E V. The ideal Z(V) can also be defined as the F-span F T of the set T. It is a consequence of the Hilbert Nullstellensatz that Z(V) is the nil radical of the ideal F[E]Sfor any S defining V (see e.g. Zariski-Samuel, [GO]: v.11: p. 16Sf). Thus Z(V) is independent of the choice of the algebraically closed field F containing F . If Ff is any extension field of F (not necessarily a subfield of F ) then the point p E P ( V F J ) is called an Ff-rational point of V if f (p) = O for every f E Z(V). It suffices to have f ( p ) = 0 for every f in a defining set of polyllornials S of V. The F-algebra F[[]/Z(V) is called the coordinate algebra F[V] of V. In dealing with a single V we shall write & = <, Z(V). Then F[(]/Z(V) = F[<]= F [ < .~. ., .&I. Since Z(V) is a homogeneous ideal (that is, contains the homogeneous parts of every f E Z(V)). ~ [ finherits ] the grading of F [ [ ] .Thus the element,^ of F[(] that are homogeneous of degree T are those of the forrn f (<) where f is a homogeneous polynomial of degree r in the E ' s lye shall now formulate Theorem 3.4.11 in these geomet,ric terms. Let V , A"(V arltl Gr(V. cl) be defined as before; so Gr(V. d) is the set of I-dimensional subspaces Fw where w is a non-zero decomposable vector contained in Ad(V). . .

.<, I,

+

Thus Gr(V. d) can be viewed as a subset of the

(.i )

-

1 dimensional projective

,

space p ( f l d ( V ) ) . Let ( e l : . . . , e,) be a base for V / F and hence for v ~ / F . The corresponding base for n d ( V ) and for A d ( v P )is (ei, A , . . Ae,, / 1 il < . . . < id n ) . We denote the set of indeterrrlinates { < Z l . . . % d / 1 5 il < . . . < id n ) by Ep SO we write F[&]for F[& l...id / 1 i l < . . . < id 5 n]. Also we define <j1...3d for arbitrary ( j l . . . j d ) as for the Pliicker coordinates. Let S be the set of polynomials

<

<

<

<

d

x ( - l ) X < i l . td-ljA$o...jXXXXjd X=O

(3.5.1)

112

111. Galois Descent and Generic Splitting Fields

<

<

where 1 5 i l < . . . < id-1 5 n,,1 jX n : 1 5 jO< . . . < jd 5 n . Then Theorem 3.4.11 states that tlle variety defined by S is Gr(VF, d) and Gr(V, d) is the set of F-rational points of this variety. Now let A be a central simple algebra of degree m over F so n = [A : F] = 1 n 2 . Then A = A f r ( D ) where D is a central division algebra of degree d and m, = rcl. If {e,, / 1 5 i , j 5 r ) are the matrix units for Al,.(D) then A = @ e j , A and IJ = e3jA = Cice J k D is a right ideal with [I:,: F] = rd2 = md where d is tlle index of A. It follows that if A 1 then A contains right ideals of dimension m over F . Conversely, if A has such a right ideal I then I is a right mod~ileof dimension m for A and hence A is isomorphic to a n m 2 dimensional subalgebra of E n d F I . Then A E EildFI and A I . Thus A -- 1 A contains m-dimensional right ideals where m = deg A. Let F be a n algebraically closed field containing F . Then AF = 1VIm(F) contains right ideals of dimension m. Let ( e l , . . . , emz): be a base for A and for AF and (eii A . . . A e,_ I 1 5 i l < . . . < i, 5 n12) he the corresponding base for Am(A) and Am(AF). Then tlie set of polynomials S of (3.5.1) with d = m, and n = m 2 defines the Grassmannian Gr(AF, m ) as a n F-variety in tlie pro,jective space PAm(AF). Moreover. we have a bijection of this variety with the set of m-dimensional subspaces of AF: If F w E Gr(Ap, m) and w = w1 . A w,, W, E A F , then the corresponding subspace of AF is C F w , . We call ilow derive some additional polynomial conditions that 7= C y ~ w , is a right idea,l of AF. Let A' denote the set of invertible elements of A. It is readily seen that F A * = A and FA*= AF. Let T be a subset of A* such that the multiplicative group generated by T contains a base for A. Then 7 is a right ideal in AE if and only if f t = f for all t E T. This coildit,ion is eyuivaleiit to FW = Fw' for w1 = ,wit A . . . A w,,t. Now the right action of A* on A given by w a defines a n action of A* on Am(A) and on Am'(AF) such that (wl A . . . A w,)n = w l a A . . . A wma. Let t E T and write

-

-

*

where t ( 3 ) , ( zE) F Then for w = Cpi,... im (eil A . . . A e,_) we have wt = C qii...i7n(eLiA . . . A eim) where

*

Heiice F w = Fwt the Pliicker coordinates (p) and (q) are proportional, that is, for all ( 2 ) and ( j )

Thus we see that the point p = F ( c ~ , , ,_(e,, A . . . A e t m ) )of PAm(AF) has the forrn F ( w l A . . . A 7~1,) where CFW,is a right ideal of AF if and only if p is a zero of the homogeneous polynomials (3.5.1) and of the homogeneous polynomials

3.5. Brauer-Severi Varieties

113

<

where1 < i l < . . . < i , , < m 2 : 1 jl < . . . < j , , < m 2 a n d t h e t ( j ) , ( , )E F. Wc shall call the F-variety defined by this system of polynoinials in F[&] for all t t T the Brauer-Severi variety V A of the algebra A. The existence of an F1-rational point on V A for F' an extension field of F is equivalent to the existence of an m,-dimensional right ideal in AFt. Moreover: the latter condition is equivalent to: F' is a splitting field for A. Hence we have Theorem 3.5.6. F' is a splitting field for A zf and only if there i s a n F1-rational point or2 the Brauer-Severi variety V A .

As an illustration of the foregoing discussion me consider the Brauer-Severi variety of a quaternion algebra A over a field F of characteristic # 2. Here we have a base ( 1 , i ,j , k ) such that i2 = al,j 2 = D l , i j = k = - j i where ~ , U FE * . Thenwehavethebase ( l A i , l A j , l A k : i A j , i A k , j A k ) forA2(A) and A 2 ( A F ) ,F an algebraica,lly closed field over F me may take T = {i; j ) . We have ( 1 A i ) i = - a ( l A i ) , ( 1 A j ) i = - ( i A k ) , (1 A k ) i = - a ( i A j ) , (i A j ) i = - a ( l A k ) , ( i A k ) i = - a 2 ( 1 A j ) , ( j A k ) i = - a ( j A k ) . The polynomials ( 3 . 5 . 5 ) determined by the choice t = i are the 15 two-rowed minors of the matrix

Also we have (1 A i ) j = (,j A k ) , ( 1 A j ) j = -,6(1 A j ) , ( 1 A k ) j = -p(l

A , j ) , (ZA j ) j = -b(l A k ) , (i A k ) j

= -,O(i

A k ) : ( j A k ) j = 8'(1 A i ) .

This leads to the polynomials that are the 15 two rowed-minors of

We must also add the Pliicker polynomial

that defines the Grassmarinian Gr(A, 2). This gives 31 equations that define the Brauer-Severi variety of the quaternion algebra.

114

111. Galois Descent and Generic Splitting Fields

3.6. Properties of Brauer-Severi Varieties One defines the Zariski topology of the ( n - 1)-dimensional projective space PVF ( V n-dimensional over F) by taking as closed sets t,he projective F varieties of PVF (cf. R. Hartshorne, [77], p. 10 or I . R . Shafarevich, [74], p. 32). If V is a subvariety of P V F we have the induced Zariski topology on V. The projective variety V is irreducible if it is not the union of two proper F-subvarieties or. equivalently, any two nonvacuous open subsets of V meet. This is the case if and only if the coordinate algebra F[V] = F [<]/Z(V) is a dornairl (Zariski-Sarnuel, loc. cit.). Here F [ [ ] = F [ [ l , .. . :En] and Z(V) is the ideal of V . As before, F[V] has t,he grading induced by that of F[[]. If V is irreducible we can form the field of fractions F ( V ) of F[V]. Of greater inlportance is the subfield F(V)o of F ( V ) of homogeneous elements of degree 0 of F ( V ) . These are the elements f E F ( V ) that have the form gh-I where g and h arc hon~ogeneouselements of F[V] of the same degree, h # 0. Direct verification shows that the set F(V)o of these elements constitutes a subfield of F ( V ) . This is called the field of ration,al functions on V. The significance of F(V)" is that its elements can be used to define F-valued functions on V. Z(V): Let f E F(V)o. Then we can write f (f) = g(f)h,(f)-' where ft = g(<) and h([) are homogeneous polynomials of the same degree in the 6's and I?(() @ Z(V). Then if p is in the non-va.cuous open subset of V defined by h(p) # 0, me can define f (p) = g(p)h(p)-l. Evidently, this is independent of the clioice of the llornogerleous coordinates. hloreover, if f (f) = gl (f) hl ( f ) ~ ' where gl and hl are homogeneous of the same degree and hl $ Z(V) then .dE)hl(E) - g~(<)h,(E) E Z(V) so f (PI = g(p)h(p)-l = gl(p)lzl(p)-l on the open subset defined by h(p)lzl(p) # 0. We now define an algebra (over F ) of functions defined on non-vacuous open subsets of V. If f l and f 2 are two such functioris defined on tjhe open subsets 01 and 02 # 8,then we regard f l = f 2 if f l (p) = f 2 (p) for p 01 n 02 (# 0 since V is irreducible). We define f l + f 2 and % i f 2 on 01 no2 by ( f + ~ f z ) ( p ) = f l ( p ) +fz(p). (flfz)(p) = f ~ ( p ) . f ~for (~) p E 01 n 02 and for n E F we define the function a by p a for all p E V. In this way we obtain an algebra over F that contains the subalgebra of functions defined as above by the f E F(V)o. This subalgebra is isomorphic to F ( V ) o . The inap of f into the function p --, f (p) is an isomorphism. La.ter on we shall need the following

ci +

-

Lemma 3.6.1. Let V be a non-zlacuows irreducible projective F-~variety;F ( V ) the field of fractions of the coordinate algebra F[V] and F(V)o the field of rational functions of V. Then F(V) is a simple transcendental exten,sion of F(V)o. Prooj. Lct ( a l , . . . .a,) be a point on V and suppose a, # 0. Then [, $ Z(V). Now let X be any nun-zero homogeneous polynomial of degree one in the ['s not contained in Z(V) (e.g. Et). If g is a homogeneous element of degree r in F[V] = F[<] then SX-T E F(V)o and g = (gX-T)Xr t F ( v ) ~ [ X ] Hence .

3.6. Properties of Brauer-Severi Varieties

115

F[V] c F ( v ) [XI ~ and F ( V ) = F(v),,(X). Now X is transcelldental over F(V)o. Otllerwise. we have elements go # 0. gl, . . . , g, E F[V] which are homogeneous = 0.X' = 0 of the same degree such that goXs+glXs-l+. . .+gs = 0. Then gois and = 0 contrary to the choice of A.

If V is a projective F-variety and F' is an extension field of F then if we assume -as we may - that F > F',we may regard V as the F'-variety defined by Z(V). \Vhen this is done we shall write VFi for V. The ideal Z(VFl) in F1[[]is the nil radical of FIZ(V).We have F1[[]= F' %F F[[] and FIZ(V) = F' C%F Z(V). Hence (3.6.2) T(V) = F [I n FIZ(V). Also it is clear from the defiilitions that

This implies that the F-subalgebra

of the coordinate algebra F1[VF1] = F 1 [ [ ] / Z ( V ~ ~is) isomorphic to the coordinate algebra F[V] = F[[]/T(V) of V. IbIoreover: since F1[[]= FIF[[]. F1[[]/Z(VF,) = F1(F[[] + Z ( V F , ) ) / Z ( V ~ / )Thus . we may regard F[V] as an ] It is clear from this that F-subalgebra of F1[VF,]such that F ' [ ~ F=/ FIFIV]. if F1[VFt]is a domain then F[V] is a domain. Hence the irreducibility of VFt implies that of V. In this case F(V)o is a subfield of the field F1(Vp)o of rational funct,ions on VF'. This is clear since an element g E F[V] which is homogeneous of degree r is also an element of F1[Vp]homogeneous of degree r in F1[Vp]. The projective space PVF where V is n-dimensional over F is itself a projective variety P V defined by the ideal 0 in F[[]. This variety is irreducible F([)o, the set of with coordinate algebra F[[] and field of rational f~nct~ions gh-I where g and h # 0 are homogeneous polynomials of the same degree. As we saw before, F([)o = F([:, . . . , [i= ti[&' and the [b are algebraically independent over F. Thus F ( P V ) o is a rational function field of transcendency degree n - 1 over F. We shall now consider the Brauer-Severi variet>yVA for A = M m ( F ) and we shall show that VA is isomorphic in a sense to be defined to PV, I/ mdimensional over F. We recall the following well-known isomorphism of the lattice of right ideals of EndFVF with the lattice of subspaces of VF: If W is a subspace of Vp let I(I.V) = {l E EndpVp I PVF c W ) and if I is a right ideal of EndFVF let V ( I ) = I V [ F ] . Then these two maps are inverses, hence are bijective between the set of minimal right ideals of EndpVF and the set of 1-dimensional subspaces of VF, which are the points of P V . If we choose a base ( v l , . . . , v,,) for V / F and replace linear transformations by their matrices relative to this base we obt,ain the following bijection: FW -. I(Fw) where w = Caivi, a, E F , and I(FW) is the m-dimensional right ideal of matrices whose columns are

111. Galois Descent and Generic Splitting Fields

116

multiples of w. Hence if (e,, 1 1 5 2 . 1 5 m) is the usual base of matrix units of nf,(F) then I ( F ~ )has the base (rl (w). . . . . r,(w)) wherc rJ (w) = atet, . TVe also have the map I (Fw) --, FLJ where J = r l (111) A . . . AT,(u'). Combining the two maps we obtain the bijection y : Fw -i Fw of the set of points of P V onto the set of points of V n f T r L ( q . We now arrange the base (e,,) for lll,(F) as (ell. ezl.. . . . e,l; el2, e22. . ; . ;. . . e,,). Then the coordinates of rl(w) relative to this base are (al, na. . . . . a,: 0. . . . , 0 ) of r2(w) are (0, . . . , O ; a l , . . . . a,,: 0, . . . , 0), etc. Hence to calculate the Pliicker coordinates of w = r l (m)A . A r,(w) we need to calculate the nz-rowed minors of the matrix

x,

It follows that the Pliicker coordinates of w determined by the base (ell, e2l. . . . .em,) for M V , ( ~are ) given by lloniogeneous polynomials (in fact, monomials) of the same degree m with coefficients in F in the coordinatcs of u. relatibe to the base ( e l , . . . . e,). NTenow consider the inverse map Fw --i F w . We can read off frorn the matrix a that the Pliicker coordinates

for 1

<

Since p

Z.J

5 m . Let

...

oi. ,-'(Fw)

0, be the open subset of Vnf,(q defined by

+,,zm+i ,....(m-l)mm+i =

=Fw =

a r and cp is bijective,

F(Cj a,ea) = F(CJar-laje,)

U Oi

= V M _ ( ~ On ).

Hence the coordinates of F w E 0, are homogeneous linear functions of the Pliicker coordinat,es of Fw and the i-tli coordinate of Fw is # 0. The foregoing discussion amounts to the fact that the varieties V n I v n ( ~ ) and P V are isomorphic. If V is a projective variety defined by a set S of homogeneous polynomials in FIE1,. . . ;<,I then a map p of V into P V (or a subvariety of P V ) is called regular if for every point p E V there exists a neighborhood U of p (= Zariski open subset containing p) such that y I U has the form JI --t ( f l (p). . . . . f, (p)) where the f, (6) are honlogeneous polynomials of the same degree conta,ined in F [El,. . . ,En] (cf. R. Hartshorne, [77]: p. 14f or I.R. Shafarevich. [74],p. 34f). If V and V' are projective F-varieties, an isomorphism of V into V' is a bijective regular map y whose inverse is regular. Then V and V' are isomorphic. I11 this sense the map y we defined above is an isomorphism of P V onto VnI,(F). For, we have homogeneous polynomials in F[[] of degree m such that

3.6. Properties of Brauer-Severi Varieties

9 is tjhe map p

--,

(fl(p), . . . , fnr(p)),M =

( )

117

hloreover, y p l is regular,

since the m subsets of V-Rf,(F) defined by (3.6.7) provide an open cover for V,ll,,z( F ) and on the subset defined by (3.6.7) p-l is the projection map sending with Plucker coordinates (p,,...,7n)into the point given by a point of Vnf7,L(F) (3.6.7). Hence we have an isomorphism of P V and VM,,,( ~ p In general, if V and V' are isomorphic then they are homeomorphic (as topological spaces relative to the Zariski topology). Hence V is irreducible if and only if V' is irreducible. For irreducible V and V' there is a weaker relation than isomorphism: birational equivalence. 4 rational ?nap V --+ 1)' is a pair (U, 9) where U is an open subset # 0 of V and y is a map of U into V' given by homogeneous polynomials of the same degree. Note that the map need not be defined on all of V. One defines equivalence for a pair of rational maps (Ul, p l ) and (U2,p 2 ) if y1 = 9 2 on Ul n U2. The map (U. 9) is a birational equivalence if there exists a rational map ( U ' , p') of V' into V such that y y ' is equivalent to l v t , and p ' p is equivalent to lv.Birationally equivalent irreducible varieties have isomorphic fields of rational functions (see Hartshorne, loc. cit. p. 24f). These considerations apply in particular to P V and VAfvn(F). We now proceed to give independent direct verifications of the results we shall require in this special case. As before, let [,,..,, , 1 5 il < . . . < i, m,2: be indeterminates and put

<

5

+

Let = (, Z(Vnl,,(F)). ViTeshow first that the are algebraically independent. Let g((1. . . . . C,) be a homogeneous polynomial in the C, such that S(C1,.. . Sm) E Z ( V A ~ , ~ ( FThe11 )). 1

<

for all the points (p,,,.,,, i,,, 1 1 il < . . . < i, 5 m 2 ) on VXI,,,(p).By (3.6.61, we have g ( a z p l a l , . . . , a:;) = 0 for all a, E I@. Since g is homogeneous t,his implies that g(a1. . . . , a,) = 0 for all a, with a,, # 0. Then g(Cl, . . . , C,) = 0. Since Z(Vl,,r,(Fj) is homogeneous this implies that the Cj are algebraically indeperldent over F. Next we observe that it follows from the form of the matrix (3.6.5) that the Pliiclier coordinate ptl ...,n, (il < . . . < i,,,) of F L ~is! 0 unless ( j - l ) m i 5 i , j m . Hence ti,...,,z E Z(VAr, (I;)) unless ( j - l ) m i i, 5 j m . LIoreover. if these conditions hold then

+ <

<

+

which implies that

for the i, as in (3.6.9). It follows that the coordinate algebra F[VM, ( F ) ] is contained in the localization .., which is a domain since the <%

p[t1..

118

111. Galois Descent and Generic Splitting Fields

are algebraically independent. Hence F[V2vfm(F)] is a donlair1 and V,v,lm(F) is onto irreducible. Aforeover, me have an isomorphism of F ( [ ) = F([1,. . . ),,[ the field of fractions of the coordinate algebra of Vn.Im(F) such that [, --i <,. Restricting to F ( [ ) o we obtain an isorrlorphism

<,cs

such that [ , [ k l --i \Ye sllall now show that if we identify the fields F ( P V ) o and F(V21,n(Fj)o with the correiporlding fields of rational functions then for f E F ( P V ) o we have

f (Fw)= (@f ) ( p ( F w ) ) (w= &e,)

(3.6.12)

(011 an open subset of P V p ) or, equivalently,

Since @ 1s an isomorphism it suffices to verify (3.6.12) for the generators [,<&l. 1 < z < m - 1, of F ( P V F ) .This holds since for f = <,[g1we have ~ ( F w=) a,akl and ( @ f ) ( p ( F w = ) ) ( ~ ~ a $ - ' ) ( a := ) ~a ~ ,al1. \Ye can apply the foregoing results to central simple algebras. If A is such an algebra then it follows from the definitions that the Brauer-Severi xrarlety VA,, = ( V A ) ~ Hence ,. if F' is a splitting field for A then ( V A ) ~=, Vni,,(F~j P ( V p ) . Hence V A is irreducible. \Ve can summarize the main results on Brauer-Severi varieties we have obtained thus far in

"

Theorem 3.6.14. ('1) The Brauer-Severi variety of a central simple algebra is Irred.ucible. (2) if A = M,(F) then F ( V A ) is ~ a rational function field of transcendency degree m - 1 over F . (3) If F' is a n extension field of F then F ( V A ) is ~ a subfield O ~ F ' ( V ~ , ,(4) ) ~The . field of fractions F(V*) of F[VA] is a simple transcendental extension of F(VA)o.

3.7. Brauer-Severi Varieties and Brauer Fields Let A be central simple of degree m over F, E a finite dimensional Galois splitting field, G = Gal E I F . n = GI = [ E : F ] . Since E is a splitting field E g~ A = EndEV where V is an m-dimensional vector space over E. Lct a , = 0 8 1 in EndEV. Then cu = { a , 1 a E G ) is a group of automorphisms in EndEV over F such that a , is a-semilinear. Evidently A = Irlv a . Now a , can be realized by a a-semilinear transformation of V. that is, there exists a bijective a-semilinear transformation u , of V such that a,P = u,Lu;'. !E EndEV. Since u , is determined up to a multiplier in E* we have u,u, = k,~,u,,,k,, E E*.Then B = C E u , 2 ( E , G ,k ) and A is ~ BO. the centralizer ( ~ n d p Vso) A Let V " = HomE(V, E ) and let 7 ~ : denote the transpose of u , in V * . The defining property of u z is that for X E V * . (u;X)(u) = a-lX(r~,u).

-

3.7. Brauer-Severi Varieties and Brauer Fields

119

E IT. which implies that u i is a-l-semilinear. Moreover, uT, is bijective. Put 6, = (u;)-'. Tlien 6, is bijective a-semilinear and 6,iiiL, = k s f i o , so CEG, "- (E. G,k-l) A. Now let ( v l , . . . , u,) be a base for VIE, (21;. . . . , v;) the dual or complementary base for V * / E (so v: (v,) = S,,). Then if

1:

-

u: is the 0-l-semilinear transformation of V* such that

The matrix li, = (u,,,) is the rnatrix of u, relative to ( v l , . . . , v,) and the rriat,rix of uz relat,ive to (v;, . . . , vh) is (a-'U,). Hence the matrix of Q, relative to (vT, . . . : uGL)is t(o-lUn)-l. \lTe now identify V* with CECi C E[E] = E[[1,. . . ,tm]: where the 6,are indeterminates, by identifying ti with the linear function 'u:. Tlien 6, ('low regarded as a a-semilinear transformation of CE(, has a unique extension t,o an automorphism q ( a ) of E(<) such that ~ ( a ) E = a. This preserves the grading in El[] so it defines an automorphism ~ ( 0 in) E([)@ ~ We have I ~ ( ~ ) ~= ~ / (( O T T)(but )~~ not q ( a ) q ( r ) = q ( a r ) ) Then Inv q(G)o is a Brauer field of dimension m - 1 as defined in Section 3.2. TVe denote this field as Fm(k-1). Now ~ ( a and ) ~ l ( a can ) ~ be characterized as certain maps of functions on VE and PVa if lie take E , as we may, to be the algebraic closure of E. First, a can be extended to an automorphism 5 of E and u, to a 5-semilinear . usual: we can ident,ify a polynomial f ( E ) E El[] transformation u, of v ~ / E As with the polynomial function on VE defined by u; = C aiw, --t f ( a l , . . . , a,). Then (17(a)f ) ( ~ ?=) 5 , f ( ~ , ~ u ; ) (3.7.3) follov,~sfor f = <,directly from the definition of the transpose. Then it holds for arbitrary f (E) E E[[] since the <, generate E[[]/E. Since E is infinite, (3.7.3) gives a characterization of q ( a )f . Similarlv. if f E E([)0 then f determines the rational function E w -i f ( a l , . . . , a,) defined on an open subset of PVE. Then q(a)0f can be characterized by the property that

holds on an open subset of PVE. n r e shall find it more convenient to work with J1fm(E)in place of EndEV. For this purpose we replace P E EndEV by its matrix L relative to the base (.cl,.. . , v,,,). Then the automorphism in ,bfm(E) corresponding to a , is L --, LT,(aL)Ull which we denote also as a,. The subalgebra of M m ( E ) corresponding to A is {L E bf,(E) / U,(aL)U;' = L, a E G). We denote this also as A. Let {e,, / 1 < L , J rn) be the usual rnatrix base for M,,(E).

<

120

111. Galois Descent and Gcneric Splitting Fields

As we showed in the last sect,ion,we have the biregular bijective map p of P V such that if u! = Cn,v,; a, E E. then onto the Brauer-Severi variety VIZI,,(~) 9 : E'ur -. ~w where LL! = r1 (UI) . . . A rm(u:) and ri(w) = C . a , e . Also we it;; fields of have tlre isonrorphism $ of the field E(PV)o onto rational functions on P V and VnITn such that (3.6.12) and (3.6.13) hold on open subsets. The a-semilinear transformation a, of Af,(E) determines a a-semilinear transformation Am(cr,) of Am(hf,(E)) such that Am(n,)(L1 A . . . A L,) = a,L1 A. . . Aa,Lm, Lz E I V ~ ~ ( Eand ) . we have the transposed map iln'(n,)* in Am(&ITIL ( E ) ) * = HornE--(ArrL(114,(E ) ) ,E ) . Then Am(a,) = (AnL(a,)*)-I is a stmilinear and An' ( ~ x , ) A ~ ~ ( a=, )Am(n,,). As before, let E[&] = EIEtl...,, I 1 i l < . . . < i m 5 m2] and let ( e l , . . . ,e,z) be a base for IZ/Im(E)/E.Then with the me can identify CEE, i,, with Am(Aln,(E))"by identifying [ where linear function on Am(Mr,(E)) such that e,, A . . . A elm --, 6(,),(3) ( i ) = ( i l , . . . i,,), ( j ) = ( j l , . . . , j,). Then Am(a,) has a unique extension to an automorphisrn X(a) of E[EP](over F) such that X(a) I E = a. We claim that X(a) stabilizes the ideal Z(V,2/I,(E)). We note first that if, as before, E is the algebraic closure of E and we define a, to be the extension of a, to a a-semilinear transforniat,ion of ,~I,(E) for 3 an extension of a to an autolnorphism of E, then for any w E Am(ilir,(E))and g = (,,...,7n we have

<

-

A

(A ( a )g) (w) = ag(Am(a,)-lu).

(3.7.5)

Since X(a) is an automorphism this holds for all g E E[Ep].Row E u is a point on V,tin,(~)if and only if w = vl A . . . A v, and Cy Ev, is a right ideal in J q n ( E ) . Since a, is a-semilinear and an automorphism of hf,(E)/F. if I is an m-dimensional right ideal of hf7,(E). then so is a,I. Hence I --i a,I is a bijection of the set of m-dimensional right ideals of 31m(2) onto itself. Then ~f Ed for w = v l A . . . A ,c is a point on VAf,(E) then so is ~ A ~ ( a , ) - l w = ~ ( a , ' u ~A . . A a,lv,) since Cy En;'v, is an m-dimensional right ideal. It now follows from (3.7.5) that X(u) stabilizes Z(V241m Then X(a) defines an autornorpllism p ( a ) in the coordinate algebra E [V,bfm = E[[p]/Z(VnI, (E)) and an automorphism ~ ( 0in) the ~ field E(V,tf,n(E))O. This automorphisln call be characterized by the property that for h E E(V,7(I,,(E))

on an open subset (depending on h) of I;M,(E). NOWp ( a ) o p ( ~ ) o= p ( a ~ ) o and p(a)O I E = a. Hence K = Inv / L ( G )is~an F-form of E(V,fI,r,(E))o. TVe wish to show that F,(k-l) r K F(Vn)O. We prove first

"

Pwoj. \Ve have Z(Vnrm(E))= Z(VA,) > EZ(V*). Since Z(VA) is prime and EIF is separable, EZ(VA) = nilrad EZ(VA (Zariski-Samuel, loc. cit., Corollary on p. 226 of v. 11). Since EZ(VA) is an ideal in E/[p defining

3.7. Brauer-Severi Varieties and Brauer Fields

121

nilrad EZ(V*) = EZ(V*). Now let f E E(V,)o. Then there exist homogeneous polynomials g ( E p ) , h ( [ p ) E E [ [ p ]of the same ( E ) ) = E Z ( V A ) such that f = g ( c p ) h ( < p ) - l degree with lz([p) $ Z(VAITn where %,,, = ELI ,m Z ( V h f m ( E ) We ) . show next that we may assunle h ( [ p ) E F[Cp].To see this let a E Gal E I F and identify a with its cxtension to E [ [ p ]fixing the [,, ,_. Then a stabilizes E Z ( V A ) and since this ideal is prime. h o ( [ p ) = a h ( [ p ) $ E Z ( V A ) and h o ( [ p ) E F [ [ p ] . ah(Er) we obtain a repreHence if we multiply g ( [ p ) and h ( [ p ) by sentation of f as g(
V,li,(E). Z(Vnl,,(=)) =

+

6,

nOEG

-

n3+,

nr=,

n,+,

We can now prove

Lemma 3.7.8. If K = Inv p(G)0 then K

F(VA)~.

Proof. SVe have M,,(E) = E 8~ A so A m ( M m ( E ) )= E 8~A m ( A ) . Now a,, A = 1 ~ SO if we choose a base ( e l , .. . , emn) for A.I,(E) to be a base for A I F then Am(a,)(e,, A . . . A etTn)= e,, A . . . A ei_. Then (Am(a,))*[,,... im = [LI...Z,n. It follou~st'hat p(a)o F ( V A ) = ~ ~ F ( v , ) .Also p(a)o 1 E = a . Then 3.7.7 implies that K = Inv p(G)o F ( V A ) ~ . •

"

SVe shall prove next

Lemma 3.7.9. K section.

" F,(k-l)

the Brauerfield defined at the beginning of the

Proof. TVe note first the conlrnutativity of the following diagram:

where [u8]: EW Then u8ui =

E u s w , [ A m ( a 8 ) ]: aaiuj,,v3 and

-i

ELL -i

E~~(a,)w Let . w = Ca,v,.

O n the other hand, c p ( E w )= ~ ( r l ( wA ). . . A r,(w)) and

111. Galois Descent and Generic Splitting Fields

122

Now Uoark(u)

- (x

Hence

[n7"( a 5 ) ] ~ ( E u 1=)

If we put

71;

=

~,,aa,e~~.

~ , ~ , t , ~ c) a(t exl k ) =

i , ~

e

i,j

u23uaaJe,kthen we have

and since this is a point in Vnr,(E), C E u i is a right ideal of M,,,(E) and hence CEvi = CEvjU;'. Then E ( v i A , . . A vd) = E ( ~ ; U ; ~A . . . A ~1kU;l). Cornparison of this with the equation for [Am( Q , ) ] ~ ( E W ) shows that (3.7.10) is commutative. The coinmutativity of (3.7.10) implies the commutativity of -1

VMrn(E)

This, (3.6.13) (with mutativity of

F

PV

replaced by E), (3.7.4) and (3.7.6) irnply the com-

The verification of this is straightforward and is left to the reader. It follows that ~ , ( k - ' ) = I i ~ vq(G)o Inv p(G)o N F(VA). We can now state the main result of this section:

Theorem 3.7.12. Th!efield F ( V A ) ofratior~al ~ ,functions on the Brauer-Severi vo,riety of A is isomorphic to a Brauer field and every Braz~erfield ,is ison~orph,ic to an F(VA)o. Proof. The first statement has been proved. To prove the second let F,(k) be a Bralier field. Then m is a multiple of the index of [k] E H 2 ( G ,E*)and we have a module V for ( E ,G, k) such that V is an m dimensional vector space over E. We identify ( E , G, k) with its image B given by the representation defined ) ~ . A is central simple and the argument by V and let A = ( E ~ ~ F VThen used above shows that F,(k-l) 2 F ( V A ) ~Hence . we obtain our result by replacing k by kpl.

3.8. Generic Splitting Fields

123

3.8. Generic Splitting Fields \;tTe recall that if K and F' are fields then a place 'P o n K to F' is a homomorphism of a subring R of K int,o F' such t,hat if a € K and a !$ R then a-' E R and P ( a - l ) = 0 (see e.g. BA 11: Section 9.7). In this case R is a valz~atlonring in K in the sense that R is a subring of K such that every element of K is either contained in R or its inverse is contained in R. Moreover. R is a local ring. Let U be the set of units of R, P the set of non-units (which is an ideal since R is local): K * the multiplicative group of units of K : R" = R n K * : P* = P n K * . Then U is a subgroup of K * ; so we have = K * / U and the subset H = P*U/U or closed under the factor group multiplication. These define an ordered group (T, H ) . If we put V = U (0) then we obtain a V-valuation cp of K by defining

r

r

r

If K and F' are extension fields of F then 'P is called an F-place if F C R and 'P is the identity on F or, equivalently. 'P is an F-algebra homomorphism. In this case the valuation cp is trivial on F, that is p ( a ) = 1 if a E F * . \;lie can now define the important concept due to Amitsur [55] of a generic splitti~lgfield of a finite dimensional central simple algebra.

Definition 3.8.2. An extension field K / F of a field F is called a generic splztttng field of a central simple algebra A / F (or the Brauer class [A]) if for any extension field F'IF. F'IF is a splitting field of A / F if and only if there exists an F-place on K to F'. Since the identity map is a place it is clear that any generic splitting field is a splitting field. \Ye shall now prove that the field F(VA)oof rational functions on the Brauer-Severi variety V A of A is a generic splitting field for A. We note first the follo~ving

Theorem 3.8.3. Let A I F be a central simple, K / F a splitting field for A / F . Suppose there exists a n F-place from K t o a field F 1 / F . T h e n F' zs a splitting j?elrl for A.

Proof. l i e have seen in Theorem 3.5.6 that F f / F is a splitting field for A if and only if V A contains an F'-rational point. Hence our result will follow from Lemma 3.8.4. Let V be a projective F-variety and let K and F' he extension fields qf F . Suppose: ( I ) V contains a K-rational point and (2) there exists a n F-place P from K t o F'. T h e n V contains a n F'-rational point.

Proof. Let ( a l , . . . , a,) be homogeneous coordinates of a K-rational point of V and let R be the valuation ring of 'P. Since R is a valuation ring and not every a , = 0 we can multiply the a , by a non-zero element of K to

111. Galois Descent and Generic Splitting Fields

124

obtain homogeneous coordinates such that one a, = 1 and all are in R. Then ( P a l , . . . . F a r ) is an F'-rational point on V. We shall also need a couple of results on fields that are rational (that is, purely transcendental) over F .

. . , [, are indetermi7zates then there exists a n FProposition 3.8.5. If place from F(E) = F ( & , . . . , [,) t o F. Moreover, if a l , . . . ,a, are arbitrary elem,ents of F then there exists a n F-place from F([) t o F that is a n extension of the homomorphism of F[E]/Finto F such that Ei --t ai, 1 5 i r .

<

Proof. It is readily seen that if P is an F-place on a field K / F to a field F1/F with valuation ring R and P1 is an F-place on F1/F to F1'/F with valuation ring R' then the composite map PIP is an F-place on K / F to F1'/Fwith valuation ring S = { a E R 1 ?(a) E R1). This remark permits us to prove the result by induction on m. If m = 1 we let F[[l](c,-,,)denote the - a l ) . This is localization of F[[1] at the complement of the prime ideal a valuation ring whose associated place is an F-place to F that extends the --i a l . We now assume the result homomorphism of F[c1]into F such that for m - 1 indeterminates. Accordingly, we have an F([l)-place from F([) = F([I) ( [ 2 , . . . , ETn)to F(E1) extending the homomorphism of F(E1)[t2, . . . , Em] to F([I) such that [, -i a,, 2 > 2. By the first result the composite of this place with an F-place from F([1) to F extending the homomorphism of F[[ ] such that El -i a1 is an F-place from F([) to F extending the homomorphism of F [[]IFsending [, -i a,, 1 5 i < m. An immediate consequence of Theorems 3.8.3 and 3.8.5 is Theorem 3.8.6. If E = F(E1,. . . [,) is rational ouer F then the canonical m a p [A] --i [AE] of Br(F) into Br(E) is injective.

Proof. Since we have an F-place from E to F, if

AE

-

1 then A

-

1.

Now let A be central simple of degree m over F . We use the notations and results of Section 3.6. In particular F[vA]is the coordinate algebra of the Brauer-Severi variety VA of A, F(VA) is its field of fractions and F(V4)o is the field of rational functions on VA. We have seen that F(VA) is a simple transcendental extension of F(VA)o and that F(VA)o is rational over F if A 1 (Theorem 3.6.14). We have shown also that if F' is an extension field of F then F(VA)Ois a subfield of F1(VA,,)o. We can now prove

-

Theorem 3.8.7. Let A be central simple over F , VA the Brauer-Severi variety of A, F(VA)o its field of rational function,^. Then,: (1) F(VA)o is a generic splitting ,field for A. (2) F(VA)o is rational over F if and only ,if A 1. (3) F(VA)~ is a regular extension of F and i f F' is any extension field of F then F 1 ( V ~ , , ) o= F' . F(VA)". (4) F1/F is a splitting field for A if and on,ly if F' . F ( V A ) is rational over F ' .

-

3.9. Properties of Brauer Fields

125

Proof. ( 1 ) We show first that F(VA)o is a splitting field. Since F ( V n ) is a simple transcendental extension of F ( V A ) O(Theorem 3.6.14 ( 4 ) ) it suffices to show -that F ( V A )is - a splitting field. Since F[V.A]= F [ [ P ] / Z ( V Ait) is clear that (. . . , hatF f [ V ~ , , = FIFIVA]and that F [ V A ]contains a non-zero homogeneous element e of degree 1. Let f E F' (VA,,)o. Then f = c~h-l where g , h E F' [ V A , , ] ~= F I F I V ~ ] are homogeneous of the same degree r . Then gePl',he-T E F I F ( V ~ ) and o ). F ' ( V A ~)o, = F' ( F ( V A , ) ) hence gh-I = (ge-') (he-')-' E F 1 ( F ( V ~ ) oHence is a composite of the subfields F' and F ( V A ) O .Since F ( V A ) ~is regular, to show that F'(VA,,)o = F' . F(VA)o it suffices to show that F 1 ( V ~ , , ) o is a free composite of F' and F(VA4)o(see p. 101). Also it suffices to do this for F' a finit,ely generated extension of F . Now tr deg F ( V A ) O / F= m - 1 = t r deg F'(VA,,)o/F'. Hence if F' is finitely genera,ted over F then t r deg F ' I F = q < oo and tr deg F ' ( V A , , ) ~ / F =t r deg F ' ( V A , , ) ~ / F ' X t r deg F 1 / F ) = ( m - l ) q . This implies that F1(Vn,,)o is a free composite of F' and F ( V A ) ~Then . F ' ( V A , , ) ~= F' . F ( V A ) O . (4) This is a,n immediate consequence of ( 3 ) and ( 1 ) .

-

3.9. Properties of Brauer Fields In this section and the next we shift the focus from fields of rational functions on a Brauer-Severi variety to the comparatively sinlpler concept of a Brauer field. By Theorem 3.7.12 the two concepts are equivalent. Again. let E / F be finite dimensional Galois, G = Gal E / F , n = IGi = [ E : F ] . Let [k]E H 2 ( G .E * ) . The element in B r ( E / F ) corresponding to [k] in the isomorphism of H2( G .E * ) with B r ( E / F ) is the similarity class of the crossed product (E.G, k). We have defined the index of [ k ]to be the index d of (E,G , k ) and we have seen (Section 3.3) that if m is a multiple of d then in the m dimensional vector space V I E we have a c~-semilineartransformation U , of V I E such that u,u, = Ic, ,u,,. Then u, determines the automorphism a , : !-i t ~ , ! u ; ~of the EndEV which is 0-semilinear. We have the F-form A of E n d ~ vdetermined by cu = {a,) and A ( E .G. k)'. \lire now revert to the procedure given in Section 3.2 for defining the Brauer field F,(k). Here

-

126

111. Galois Dcscent and Generic Splitting Fields

we identif) V with the subspace Cy EC, of E(6) = E((1.. . . .Em) (rather than IT* with CE, as in Section 3.7) and we extend u , to a r-semilinear automorphism ~ ( a of ) E([). Let ~ ( c r denote )~ ~ ( a 1 )E ( < ) o Then we define F,,(L) = Inv v ( G ) ~where v(G)0 = { ~ ( r I )r ~E G) is a group of automorpllislns of E ( [ ) o and, as before, we call F,,(k) the Brauer field of dinlension m - 1 defined by [k]. This is an F-form of E([)o. We have shown in Section 3.7 that Fm(kP1)E F(VA).Replacing k-I by k we see that Fm(k) 2 F(VAo) itnd since AO ( E . G, k) we have the following immediate consequence of Theorem 3.8.7. (1) and (2).

-

Theorem 3.9.1. Let [k] E H 2 ( G ,E*) and let m be a multiple of the index of [ k ] . Then: (1) F,,(k) is a generic splitting field for ( E , G , k): (2) F,(k) is ratzonnl o v e ~F if and only if k 1.

-

\Ve recall the corlcepts of inflation and restriction for factor sets and crossed products (Section 2.13). For the first of these we suppose E/F is Galois with Galois group G and k = {k,,, / 5 , T E G ) is a factor set for G with values in E* and E/F is a Galois extension of F containing E/F as subfield. Then we have the inflation inf k which is the factor set for G into E' defined by k,,, = kc, where a and T are the restrictions of a and r respectively to E. We showed in Theorem 2.13.8 that (E.G, k) ( E . G, k). The proof was obtained by considering a module V for ( E , G. k) and using it to construct a nlodule V for (E,G, k) as V = E @JE V. Then we showed that the centralizer of ( E . G. k) in EndFV is isonlorphic to the centralizer of (E,G , k) in EndFV. Using the present notations we denote these centralizers as 2 and A respectively so we have A % A. In the proof of 2.13.8 we used the irreducible (E,G , k) module V. However, no use was made of the irreducibility of V so the argurlient applies to any (E,G, k) module V Then [V : E] = rn = [V : E] and m can be taken to be any multiple of the index of (E, G, k ) (= index of (E.G , k)) and we have the Brauer fields Fm(k) and F,(k). Since these are isomorphic to F(VAo)oand F(V40)o respectively it is clear that F,(k) E F,(k). We state this inflation theorem for Brauer fields as

-

Theorem 3.9.2. Let E / F be Galois uiith Galois group G , E/F a Galois szibfield of E / F with Galois group G and let be a factor set of G with values i n E" . Let k = Inf k . T h e n F,, (k) F,(k).

"

IVe prove next the analogue for Brauer fields of the theorern on restriction fol crossed products (2.13.16). This result asserted that if F' is any extension field of F then (E,G,k)r;l (E1.G'. k') where E' is the composite EF'. G' = Gal E'/F1 and the factor set k' is defined by restriction: k&,,,, = k,tE r , l p It is clear that the index of ( E , G, k ) ~is) a divisor of the index of (E.G. k). Hence if F, (k) is defined then so is F,(K1).The following theorem gives the relation between these fields.

-

3.9. Properties of Brauer Fields

127

Theorem 3.9.3. Let F' be a n extension field of F , E' = EF', G' = Gal E'IF' and let the factor set k' be definRed b y restriction, from k . Th,en F A ( k l ) = F' . F,,( k ).

Prooj. Since we are assuming that F,,(k) is defined we have an ( E .G. k ) rnodule V with [V : E ] = m. Suppose first that E' = E , or equivalently, F' is a subfield of E . Then G' is a subgroup of G and ( E ,G', k') is a subalgebra of ( E .G . k ) whose center is F' since G' = Gal E / F 1 . Now ( E .GI. k') has a representation in V / F 1 and the Brauer field Fn,~(lcl)is Inv q(G1)owhere v(G)o is the group of automorphisms ~ ( a )a~ E, G , of E(<)o defined by the 21,. Since F,(k) = Inv q(G)o me have F m ~ ( k 1>) F1F,(k) On the other hand, the olily ~ ( 0that ) ~fix the elements of F' are the q(a)o with a E G'. Hence FIF,(k) = F A ( k l )by the Galois corresponce applied to E(<)o as Galois extensioli of F,,,(k). Si~ice[F' : E ] < m. FIF,,(k) = F . F,(k). Hence FA ( k ' ) = F' . F, ( k ) .

"

Suppose next that E n F' = F . Then E = FE' E % F F' and ( E ' .G', k ' ) ( E .G , k ) p has a representatioli on V' = V p and [V' : E'] = m. The field of homogeneous rational functions of degree 0 in E 1 ( < l.,. . : Ern) is E'( F' 8~E ( [ ) o and ~ , ( a ' )I ~E ( O o = ~ ( 0 )Hence ~ . F A ( k l ) > F' . F m ( k ) . To prove F,,(k1) = F' . F,(k) it suffices to show that E' F' . Fm(k) = E' E F F ~ A ( k ) = E1([)o. Now E' aF,F' . F m ( k ) is a subdomain of E ' ( [ ) o ,helice a field, and this con(F' EF F,(k)) = E' @IE ( E 8~F,(k)) = E' % E E(<)o. The tains E' argument used to show that El([)" = F' . E ( O o can be used to show that E' @E E ( [ ) o= E' . E(<)o= El(<)@Hence E' [@F' (F' . F,,(k)) = E1(<)oand F' . F,,(k) = F A ( k l ) .

"

<

"

8

~

1

~1

The proof can be completed by combining the two cases uTehave considered. as iii the proof of 2.13.16. We leave the details to the reader.

-

Now (F,G . k ) ~ ! ( F ' .GI. k') so F' is a splitting field for ( E .G. k ) if and 1. By Theorem 3.8.7 (2), this is the case if and only if FA(k1)is only if k' rational over F'. Hence we have the following consequence of Theorem 3.9.3. Theorem 3.9.4. F 1 / F is a splitting field for ( E ,G , k ) zf and only if F1.F,(k) is rational over F'.

We shall obtain next a relation between Brauer fields F,(k) and Fmi ( k ) for different m and m ' . We rnay assume m 2 m'. Then we have the following

128

111. Galois Descent and Generic Splitting Fields

Theorem 3.9.5 (Roquette [63]). If F,(k) an,d F,t(k) are defined and m > nz' t h e n F , ( k )is isomorphic t o a rational function field of transcendency degree m - m' over F,! ( k ) .

Proof. LLTehave the ( E ,G. k)-module V = Cy E[, such that V = V' @ V" where V ' = E[;. V" = E[;' where (ti,. . . , [ A , ,.;[ . . . , [ k t , ) = ( c l , .. . , E m ) . Then we have the Brauer fields F,(k), F,,,((k) and Frn/r(k). The first is the subfield Inv v(G)o where v(G)o is the group of automorphisms of E([)O defined above. Let v 1 ( o )= ~ v(a)o I E([')o where E ( [ ' ) = E(<:,. . . , [k,). Then F,, ( k ) = Inv V ' ( G )and ~ we have a similar definition of F,,l/(k) = Inv v 1 ' ( G ) ~ Now ( E .G. k)l;;,,(k) (EFrnt(k).GI, k'). Since F,tl(k) is the Brauer field of dimension rn" - 1 of (E. G , k ) ,Frn,( 5 ). F,~J ( k ) is the Brauer field of dimension m" - 1 of ( E . G ,k)F,,,(k)by Theorem 3.9.3. On the other hand. since F,,, ( k ) is a splitting field for ( E ,G. k ) . (EF,) ( k ) ,G', k ' ) 1 in Br(F,,/ ( k ) ) . Hence by Theorem 3.9.4. Frn,( k ). Frnt,( k ) is rational over F,,, ( k ) of transcendency degree m" - 1 = m - m' - 1. Our result will therefore follow from the following

x;"

~ 7 '

N

-

Lemma 3.9.6. Frn(k)is a simple transcendental extension of K = F,((k) F,,, ( k ).

.

Proof. Consider EIC. This is the subfield of E ( [ ) " = EF,(k) generated b y EF,,,(k) = E ( [ ' ) o and EF,,,(k) = E ( [ " ) o . Since these subfields are stabilized by rl(G)o so is E K . Moreover, the map a --t ~ ' ( a=) v(a)o I E K is a monornorphism. Hence lv*(G)1= n = IG/ and [ E K : Inv v * ( G ) ]= n. Also K c Inv r/*(G)and [ E K : K ] 5 [ E : F ] = n. Hence K = Inv rlL(G). \Ve show next that EFrn(k)= E ( [ ) o is a simple transcendental extension of E K . We have E ( O o = E(C1... . ,(,,-1). 5, = [,[&', EF,/(k) = E([')o = E(<:. . . . ,
<;

<

<

and since u,[,/ E CEE;, (u,[,l)[,; E E ([')o.Similarly. (u,[,)[;l E E ([")o and hence (rl(a)o<)<-l E EK. Now put = (v(a)o<)<-l.Then ( , = <,(q(a)o<,) and hence there = (rl(a)oX)X-l. Then ,u = <Ap' E Inv v(G)o = exists a X E E K such that F,(k) and E ( [ ) o = E K ( p ) . Since E([)oEF,,(k) E @ F Frn(k).the equality E ( [ ) o= E K ( p ) ,u E F m ( k ) implies that F,(k) = K ( p ) . It is clear also that p is transcendental over K .

<,

<,

3.9. Properties of Braucr Fields

129

Theorem 3.9.5 can also be formulated in terms of fields of rational functions of Brauer-Severi varieties.

>

Theorem 3.9.5'. Let D be a central division algebra of degree d and let s s f . Then F ( V n f s ( D ) )iso isom,orphic to a rational function field of transcendency degree d ( s - s') over F(VAr,,( D ) ) o .

Proof. Let the notation be as in the proof of Theorem 3.9.5 and let A = Inv a where a = { a , / c E G ) , where a , is the automorphisrn ! --t u,!u;' of EndEV Then A is the centralizer of Eu, N (E, G , k ) in EndFV. Moreover, 7n = [V : E ] = sd where d is the index of ( E .G , k ) and s is the number of irreducible components of V as ( E .G , k)-module. Then A % ~ z . ~ , ( D O ) (if D is a division algebra in [ ( E G , , k ) ] ) .Similarly, if we replace V by V' we 1WS,(D0)where [V' : E ] = m' = s'd and A' is the centralizer of obtain A' C E I L ,in EndEVr. Now F(VAo) S F,,(K) and F(VA,o 2 F,(Kf) and since A0 % AIs(D) and A'O = M s , ( D ) F(VA418(D)) , is isomorphic to a rational function field of transcendency degree m - m' = d(s - s') over F ( V M , , ( D ) ) .

xuEG

"

As in Section 3.2, let A denote the group of automorpllisms of E(E)o defined by PG!,(E): If I = ( t i j ) E G!,(E) then !defines the automorphism q(k) of E([)such t>hat ~ ( l( )E lf, . . . , Em) = f (C&lEz,.. . -CjlrnEz). The restliction q(QO of q(!) t o E ( O o is an automorphisin and the q(!)o. E E G!,,(E), form a group A that is isomorphic to PG!,(E) We shall now determine the group of A-automorphisms of the Brauer field F,(k) as defined on p. 94. We shall call these h e a r automorphisms of F,(k). As shown in Section 3.2 (p. 99), this group is isomorphic to the subgroup of A consisting o , E G. It is readily seen that this subof the X that centralize the ~ ( a ) 0 group is isomorphic to the subgroup of PG!,(E) of cosets E*! such that (E"!)(E*u,) = (E"u,)(E"!),a E G. We have seen also that if A is the form of EndEb' determined by a = { a , ( a E G ) . a, : 1 -t 7 1 , E ~ ; ~ . then the automorphlsm group Aut A is isomorphic to the same subgroup of PGI,,(E). Thus the group of linear automorphisms of F,(k) is isomorphic to Aut A. Xow if ( E .G , k) -MT(D)where D is a division algebra. then A r M s ( D O ) where m = sd and d is the index of D. Since the automorphisms of A are inner we h a w the following Theorem 3.9.6. Let m = sd where d is the degree of the division algebra D i n [(AG . , k ) ] .Then the group of linear automorphisms of the Brauer field F m ( k ) is isomorphic to M ~ ( D O ) * / Fwhere * Ms(DO)*is the multiplicative grou,p of invertible elements of Als (DO).

130

111. Galois Descent and Generic Splitting Fields

3.10. Central Simple Algebras Split by a Brauer Field As before, Irt EIF be finite dimensional Galois, G = Gal EIF, jGI = [E: F] = n and let [k] E H z (G, E * ) . Let m be a inultiple of the index of [k] = index (E,G, k). Then we have the Brauer field F,(k). \Ve recall the definition: We have a rl~oduleV for ( E . G, k) such that [I' : E] = m. Then for a E G we hare a a-semilinear transfornlation u, such that u,u, = k,,,~,,. If ( e l , . . . , e,,,) is a base for VIE we can identify V with EEi C E((1.. . . : 6),; 6, iiidrtermi~iates,by identifying e , with <,, 1 5 5 m. Then u, can be regarded as a a-semilinear transformation of E[,. This has a unique extension to an automorphism ~ ( aof) E(6) = E ( & , . . . ,Em) such that q ( a ) I E = a . Then q(u) stabilizes E(<)o the subfield of E([) of homogeneous rational functions of degree 0. Then if ~ ( 0=) ~~ ( a / )E(<)o,r j ( ~ ) ~ q=( ~v )( ~a ~ ) ~ and F , ( k ) = Inv q(G)O. We have EF,(k) = E 8~F,(k) = E(<)o and F,(k) is a generic split,t,ing field for (E.G, k)O and hence for ( E : G. k). We now consider Br(F,(k)/F), the subgroup of Br(F) of algebra classes split by F,,(k). We have [(E:G, k)] E Br(F,(k)/E) and hence the cyclic group ( [ ( E .G, k)]) C Br(F,(k)/E). The order of this cyclic group is the exponent e of [k] = exponent of ( E , G, k). The main result of this section is

Cy

Cy

Theorem 3.10.1 (Amitsur [ 5 5 ] , Roquette [63]). The Brauer group

Proof (Roqwette). Let B be a central simple algebra split by F,(k). Since E is a splitting field for (E,G , k) and F, (k) is a generic splitting field for (E,G , k) we have an F'-place on F,(k) to E . Since F,(k) is a splitting field for B so is E . Thus we have Br(F,(k)/F) C B r ( E / F ) . (3.10.1) Since B r ( E / F ) = {[(E.G , a)] I [a] E H 2(G. E')), the theorem will follow by sllowing that

Since E E F, ( h ) = E(<)o and Gal E(<)/F,(k) = I ~ ( G ) by " , the restriction t lleor em on crosscd products (2.13.16). we have (3.10.3)

(E, G, a ) ~ , , ( k )" (E(6)o:77(G)o:a') as algebras over F,(k)

Then (E.G. a ) F,, homomorpl-lism

(A)

where a' is defined by

-

-

aLi(a)o.q(r)o -

(3.10.4)

'g.7.

1 if and only if [a'] = I. The map [a]

Res : H'(G, E*)+ H ~ ( ~ ( GE([)G). )~.

--,

[a'] is a (3.10.5)

131

3.10. Central Simple Algebras Split by a Brauer Field It follows that [ ( E ,G, a ) ] t Br(F,,(k)/F)

*

[a] t ker(Res (H'(G. E*)+ H ~ ( ~ ( G )E([);)) ".

(3.10.6)

It is natural to replace q(G)O by G and consider E ( < ) i as a G-group via o f = ~ ( r r )f". Then H2(q(G)0,E([)G) becomes H 2 ( G .E([)G) and tlie restric:t,ion honlomorphism is Res H 2 ( G .E * ) + H 2 ( G ,E([);I). Our result will follow if we can show that jker(Res ( H 2 ( G .E*)+ H 2 ( G ,E(<)i)l5

C.

(3.10.7)

<

e, and For. by the commutativity of (2.13.19), this inlplies Br(F, ( k ) / F e, it will follow that' since B r ( F , , ( k ) / F ) > ([(E.G , k)]) so IBr(F, ( k ) / F ) I Br(F,(k)/F)I = e arid Br(F,,(k)/E) = ([(E,G, k)]). The proof of (3.10.7) will he based on actions of G on certain rnmiltiplicative groups associated with E(t) that we proceed to define. Let A{ be the niultiplicative group of homogeneous rational expressions f 0 in E ( 0 . This contains E([)G, the inultiplicative group of non-zero elements of E([)o, and it also contains the submonoid H of non-zero homogeneous polynomials. Tlie submonoid II is factorial. Hence every element of M can be written in the form f(E) = PI(<)^' . , . P T ( [ ) ~ ~ (3.10.8)

>

where n E E'. the p,(<) are irreducible (or prime) elements of H , and the e, E Z. Such a factorization is unique up to multipliers in E*.

Put D=>Lf/E*.

Do=E([)h/E*

(3.10.9)

E 12.f we define the degree deg f ( < ) E W = so Do is a subgroup of D . If f (0 deg f ([). Thcn Do is the subgroup of D of elements of degree 0. Thus we have the exact sequence l-+Do'iD+Z+l (3.10.10) where Z is the additive group of integers and D deg f ( t ) . LI7e also have the exact sequence

+

Z is the map f ( O E * --,

We shall define next actions of G on the terms of (3.10.10) and (3.10.11). On E* this will be the given Galois action and on Z , G mill act trivially. As usual, let the cailonical generators u,. a E G, be regarded as a-semilinear transformations of V = Et,. Then 21, extends to a usemilinear automorpliism ~ ( a of) E ( < ) . If f is honiogeneous of degree 1- then r l ( o ) q ( ~ ) 1 7 ( a ~f ) = p 1 kL,7 f . Then q ( a ) q ( r ) q ( a r ) - l ( E * , f )= E*f . Hence we have an action of G on D defined by a ( E * , f )= E q ( q ( a )f ) . Do is a G-subgroup of D under this action and u preserves degrees. Hence (3.10.10) is an exact sequence of G-groups. Also (3.10.11) is an exact sequerlce of G-groups sincc t'he action in Ex is the re~trict~ion of that in E([)G. These exact sequences give rise to long exact sequences of coliomology groups a portion of which me shall

xy

132

111. Galois Descent and Generic Splitting Fields

require. As a preliminary for determining these we list the following H O and H 1 terms: I . H O ( G . E * ) = E~~ = F * . 2. H Y G . E (0;) = F, (k)' (by definition o f the Brauer field). ) = a, f ( E ) , a, E E x . u E G ) . W e 3. H O ( GD , ) = DG = { f ( ( ) E x I ~ ( af (() call the f ( E ) satisfying these conditions G-semr-znvarzants o f k1. 4. H 1 ( G ,E x ) = 1 , by Noether's equations. 5. H 1 ( G , E ( ( ) $ = ) 1. by Noether's equations. Less immediate is

Proof. D is a direct product o f the cyclic groups ( E ' p ) where p is a prime hoinogeneous polyiiomial. Let D, denote the product of the ( E X ( a p ) )for all u t G . T h e n D, is a G-subgroup o f D and D is the direct product o f the D p . Hence H 1 ( G ,D ) = H 1 ( G ,D p ) SO it suffices t o prove that H 1 ( G .D p ) = I. Let G , be the stabilizer o f E X p . T h e n Dp is the induced G-group 1 n d z p ( E x p )induced from the G-group ( E * p ) . Hence. b y Shapiro's lemma H 1 ( G ,D,) N H 1 ( G ,( E * p ) )(see K.S. Brown, Cohomology of Groups. Springer Verlag. 1982. p. 73). Now G , acts trivially on ( E * p ) .Hence ) Hom(G,. ( E ' p ) ) and since G p is finite and ( E * p )is torsion H1(G,. ( E X p ) = free, Hom(G,, ( E * p ) )= 1. Hence H 1 ( G .D,) = 1 and H 1 ( G .D ) = 1.

n

-

T h e long exact cohomology sequence for (3.10.11) now gives the exactness of 1

+ F*

F, ( k ) * + H O (G,D ~ +) 1.

(3.10.12)

T h e long exact sequence o f (3.10.10) and (3.10.13) give the exactness o f

T h e image o f DG under DG + Z is the subgroup of Z consistirlg o f the integers that are degrees o f semi-invariant elements o f M . Let this be Ze'. T h e n we claim that 0 < e' 5 e where e is the exponent o f ( E ,G , k ) . This follows from the following

Lemma 3.10.15. If e is the ezpon,ent of ( E ,G , k ) then there exists a semi/n~uarlnntin M of degree e. Proof. Choose z # 0 in V = C y EC,. T h e n u,z E V and hence z, = (TL,z)z-' E E ( [ ) 0 .W e have u,u,z = k,,,u,,z; hence

3.10. Central Simple Algebras Split by a Brauer Field

Since k e

N

1 there exist

t,

133

E E* such that

Hence if we put w, = z;lg1 then w, E E(<)o and w,(uu;,)Iu&! = 1. Hence we haxre a q E E([)G such that U I , = q ( a q ) - l . Now put f = zeq. Since z is horllogerleous of degree 1 and q of degree 0, f is homogeneous of degree e. We have

( d a ) f) f

= (u,~)~(aq)z-~q-' =

( ( I L , z ) z - ~ ) ~ (=~z ~: w) ;~~ -=~e,.

Hence f is a semi-invariant of degree e. By the exactness of (3.10.14),we have

where e' is the smallest positive degree of semi-invariants of M . Then

Next we consider again the long exact sequences defined by (3.10.11). Using (3.10.14) u7eobtain the exact sequence and since we have H 1 ( G ,Do) Z / Z e 1 we have an exact sequence

"

1

Z/Zel

+ H 2 ( G .E * ) + H 2 ( G ,E ( 6 ) : ) .

(3.10.18)

Since. by definition the map H 2 ( G ,E*) + H 2 (G, E ( [ ) $ )in the long exact sequence is the restriction map, (3.10.18) implies that ker Res H ' ( G , E * ) + H 2 ( G ,E(<)GI = el. Since e1

(3.10.19)

< e we have (3.10.7) which implies the theorem.

Also the proof shows that e' = e and since e' is the smallest positive degree of G-semi-invariants under the action of G by a f = q ( a )f , we obtain

Theorem 3.10.20 (Roquette [63]). T h e exponent of ( E ,G , k ) is the smallest positi,ue deg'ree of serni-invariant homogeneous functions i n F ( 0 . If we recall the connection between Brauer fields and fields of rational functions on a Brauer-Severi varieties (Theorem 3.7.12) we obtain the follo~ving result.

"

Theorem 3.10.21. I f F ( V A , ) O F ( V A , ) ~for central simple algebras Al and A2 t h e n [All and [A'] generate the same cyclic subgroup of the Brauer group. An interesting question that was first raised by Amitsur for central division algebras is: Does the converse of Theorem 3.10.21 hold? The alternate form of this in terms of Brauer fields is: Is F,(k) F,(kt) for every t prime to the exponent. Amitsur in [56]showed that this has an affirmative answer if the

"

134

111. Galois Descent and Generic Splitting Fields

divisioli algebra D in ( E . G, k) is cyclic. Roquette (1641) gave another proof of Arnitsur's theorem and proved a number of other res~iltsthat give further evidence that the isorriorphisrn theorern is valid. \Ye shall now give two of his easier results. F m ( k E )zf ( L , e ) = 1 a n d m Theorem 3.10.22. F,(k) ~ n d e xa n d exponent of [k] respect~uely(or of (E.G. k)).

> d , d and e the

Proof. F,,(k) is a splitting field of (E,G. k) and hence of (E,G. k t ) . Then L = F,(k) . Fm(k" is rational of transcendency degree m - 1 over F,(kp) by Theorein 3.8.7. Since is prime to e , (E;G, k) -- (E, G , ke") for some O'. Hence the roles of (E,G, k) and (E.G, k') can be reversed to show that Fm(k) . F,, ( k ' ) is rational of transcendency degree m - 1 over Fm(k). Let t be an indet,erminate then L(t) is of transcendency degree m over F,,(k) and over ~,(k"). Then, by Theorem 3.9.5, L ( t ) Fz,(k) and L(t) F2,(kt). Hence F2,(k) ,?2,(ke) and this holds for any m divisible by d . In particular, F2d(k) 2 ~ ~ ~ ( kNOW ' ) .let ,rn> d. Since m is divisible by d , m 2d. Then, by 3.8.8: F,,,(k) is a rational function field of transcendency degree m - 2d over F2d(k).Similarly, Fm(kE)is a rational function field of transcendency degree 7n - 2d over ~ ~ ~ Hence ( k ~ Fm ) (k) . % F,(kF).

"

"

>

T;lTeshall prove next that for ariy mm,there is a mono~norphismof F,(k) into ~ , ( k ' ) a i d a monomorphism of F,(k" into F,,(k). The proof will be based oil the following theorem on fields also due to Roquette ([64]).

Theorem 3.10.23. Let F be an, in$n,ite field, E a n d K elctensior~fields of finite t r a n s c e n d e n c y degree o v e r F . Suppose EIF c a n be imbedded i n a ration a l f u n c t i o n field of finite tran,scen,dency de,gree over K and ti- deg K I F t r deg E I F . T h e n EIF c a n be imbedded i n K I F .

>

Proof. ltTemay assume E/F is a subfield of K ( [ ) / F where K ( < ) = K(E1, . . . -6,) and the <,are indeterminates. Then we have the following diagram of field inclusions:

and wr have to prove the existence of a moiioinorphism of (assuming that t r deg K I F 2 tr deg E I F ) .

EIF into

K/F

3.10. Central Simple Algebras Split by a Brauer Field

135

l i e shall first reduce the proof to the case in which the following two supplementary conditions hold: (i) t r clcg E/F = t r deg K I F (ii) t r deg K ( < E(E) ) = 0.

>

Suppose t = t r deg K / F - tr deg E/F > 0. Then tr deg E ( K ) / F tr deg KlF = f t r deg E/F. Hence t r deg E ( K ) / E t and so K contains t elcmeiits ~ 1 ,. .. , that are algebraically independent over E . Then tr deg E(?jl.. . . . v t ) / E = t r deg K I F and an imbedding of E(vl. . . . . ztt)/F in K / F gives an imbedding of E/F in K / F . Hence we nidy assume (i). l f e put m = tr deg E/F = t r deg K / F .

>

+

rlt

Nou- suppose tr deg K (<)/E(E) = f . Then

r

+m =

tr deg K([)/K

+

= t r deg K(<)/E([) =f

+

tr deg K I F = t r deg K ( < ) / F

+

tr deg E ( < ) / E

+

tr deg EIF

t r deg E ( < ) / E + m .

Herice

tr deg E(E)/E = r

-

f

and tr deg E ( [ ) / E = r ++ t r deg K(<)/E([) = 0. Now suppose f > 0 and let v l , . . . . v f be elements of K that constitute a transcendency base for K(<)/E([) and let (Ef + I . . . . , [,) be a transcendency base for E([)/E. Put

Then K(C1'. . . ; C,.) = K(E) since the v, E K . Hence the C's constitute a transcendelicy base for K(C)/K = K([)/K. since K ( < ) = > E we may replace the ['s by the ('s. Now v l . . . . ,ad are algeK(<) braically independent over E([). Hence . . . : Cf are algebraically independent over E(6). Also <S+l,.. . , C, are algebraically independent over E . Hence C1.. . . , <, are algebraically independent over E. Thus tr deg E(C)/E= and tr deg K ( ( ) / E ( < ) = 0 and so we have (ii) on replacing the ['s by t,he ['s. lVe rlo~t~ assume (i) arid (ii). Let c = ( e l , . . . c,), c, E F, and let PCbe a K-place on K ( < ) to K that extends the horriornorphism of K[[] onto K such that [, c,, 1 5 i 5 r ; (Proposition 3.8.5). Let R, be the valuation ring of PCP,, PCits maximal ideal. The restriction of PCto E is an F-place on E/F to K I F u-liicli will be a inonornorphism of EIF into K / F if E c R,, the valuation ring of PC.or, equivalently, E n P, = 0 for the maximal ideal P, of R,. Since E n R, is a valuation ring in E its field of fractions is E and its irnage PC(E) P , ( E n R,) is a subfield of K I F . Now PC1 E will be a monomorphism if

-

-

t r deg E/F = tr deg ( E n R,) = tr deg P,(E)

136

111. Galois Descent and Generic Splitting Fields

(see Zariski-Samuel I, Theorem 29 on p. 101).Since tr deg E I F = tr deg K I F this is the case if tr deg K I F = t r deg P,(E) and hence if I< is algebraic over P c ( E ) .Thus our result will follow if we can show that the c, can be chosen in F so that K is algebraic over P c ( E ) . Now let a E K . By (ii) a is algebraic over E(E). Hence we have a relation f the form

where the fi

( E ) E E[E]and f n ( 6 ) # 0. We have

where the jk

> 0 and a,(3)E E . Also

) E K [ [ ]and every hi(j ) ( E ) f 0. Now put where ,9,(j) ( 6 ), h z ( 3(t)

Then h(E) is a non-zero element of K [ [ ] .Hence the c = ( e l . .. . . c,). c, E F . such that h(c) f 0 is an open subset of F(') and for any c in this set every E RC and f n ( ( ) @ P i . Then f,(E) E Rc and P , f , ( t ) = Z P C ( a , ( , l ) .~. l. c$ E IP,(E) and PCf,([) # 0. Hence we can apply PC to (3.9.15) to show that a is algebraic over P C ( E ) . Now let (a('), . . . . a ( t ) )be a transcendency base for K I F . For each a(k)we have a open subset o(" of F(') such that if c E o ( ~ then ) is algebraic . for c E 0 = no(" , a(" is algebraic over P c ( E ) .It follows over P C ( E ) Hence that if c E 0 then K is algebraic over P c ( E ) and hence as we saw above, P, gives a monomorphism of E I F into K I P . We can now prove

Theorem 3.10.28. Let ( E ,G! k ) have exponent e and index d and let 17-1 = ds. Then for any C, the Brauer field F,(kt) can be imbedded in the Brauer field F m ( k ) .Hence if ( e ,C) = 1 then each of F,(k) and ~,(k" can be imbedded in th,e oth.er.

Proof. The result is clear if F is finite. Hence we assume F infinite. Since F,(k) is a splitting field for (E,G, k e ) .F,(k) . F,(ke) is rational over F,(k). Thus we have an imbedding of Fm ( k e ) in a rational function field over F, ( k ) . Since all of these fields are of finite transcendency degree over F we can apply Theorem 3.10.23 to conclude that F,(ke) can be imbedded in F,(k). Evidently, it follows from Theorem 3.10.21 that if A1 and A2 are central simple with the same splitting fields then [A'] and [Az] generate the same subgroup of the Brauer group. On the other hand, the following example,

3.10. Central Simple Algebras Split by a Brauer Field

137

com~nunicatedby Burton Fein, shows that this conclusiorl may not hold if Al and A2 have the same finite dirnerlsiorlal splitting fields (or, equivalently, the same algebraic splitting fields). Let the base field F = Q . The11 we can apply Hasse's determination of Br(Q). We need t o consider first Br(Q,) where Q p is the field of p-adic numbers. It is known that Br(Q,) is isomorphic to the additive group Q T / Z T of rational numbers mod 1. We have a canonical isomorphism ,Op of Br(Qp) onto Q + / Z + (see, e.g. BAII p. 610f). Now consider Q and its absolute values. There are the usual (archimedean) absolute value and the p-adic absolute va111es / 1, for every rational prime JI. The corresponding completions are R (or Q,) and the Q,. Now let [A] E Br(Q). Then for any rational prime p and for x we have the element [A], = [AQp]and [A], = [AR]. We define invp[A] = &([A,]) and inv,[A] = 1 or according as AR -- 1 or AR 7: 1. These elements of Q f /Zt are alrnost all 0 (all but a finite number) and satisfy Artin's reciprocity law inv, [A] = 0. Conversely, given any map of the set of primes plus no into the additive group then there exists an [A] in Br(Q) of ratioilals mod 1 satisfying these ~ondit~ions having this map into Q + / Z + as its map. If we define the product of maps by component-wise addition we obtain a n isomorphism of Br(Q), wit,h the group of these maps. A complete proof of Hasse's t'heorem on this isomorpllisln is given in Deuring's [35],p. 114f, and a proof assunling the requisite arithmetic facts is given in R.S.Pierce, [X2], p. 357f. We shall also need the condition that an algebraic number field K / F be a. splittilig field for A. For any rational prime p or oo let Kg be the completion of any valuation of K extending 1, or / 1. Then K is a splitting field for A if and only if [1(6 : Q,] or [Kg : R] is a multiple of the index of AQp or AR. (See Deuring's [33],p. 113.) We can now give Fein's example. Let p, q, r be distinct primes. Let [Al] E Brn(Q) be defined by inv,[Al] = 215 = inv,[Al], inv,[A1] = 115 and [Az]by inv,[Az] = 215 = inv,[Az], irv,[A2] = 115. Then invp[AlIt = 2t/5(rnod 1) so if [Allt = [A21 then t r 1. This contradicts invq[AlIt = 2t/5 and invq[A2]= 115. O n the other hand, the criterion quoted implies that, since the index of [-411 and of [A21 = 5 (since the indices of [Ailp, [A,Iq and [A,], are 5) then [All and [Aa] have the same finite dimensional splitting fields.

111. Galois Descent and Generic Splitting Fields

138

3.11. Norm Hypersurface of a Central Simple Algebra 111 thi.; section we define another projective variety associated with a cent ~ a siniple l algebra A. the norm hypersurface. Let ( u l . . . , u,). be a base for A I F . El.. . . . E,, indeterrninates. As in Section 1.6. we let 2 = CIEr, E AF(<) and let m, (A) = Am - 7-1 (3.11.1) ... (-l)m~ (<)m

+ +

be the minimum polynonlial of z in AF(0. Then T,([) is a homogeneous polynomial of degree i in the <'s. We put n(n:) = r m ( x ) and we defined the reduced norm n ( a ) for a = Cuiui E A by n ( a ) = 7 , ( a l , . . . :urn). This is independent of the choice of the base and is unchanged if we pass from A t o AE: E a n extension field of F . hloreover: n(A1 - x ) = m,(A). If F is a n algebraically closed extension field of F then we define the (projecti,oe) n o m hypersurface or hypers~~rfuce of singlrlar elements SAas t,he subsct of P A F consisting of the points p = Fa such that n ( a ) = 0. If A = M,(F) then we car1 use t,lle usual matrix base (ei3 / 1 5 i. j 5 m ) and Then n ( x ) = det x ? n: = C<,je,j and the corresponding indeterminates ti,?. a such that n ( a ) = 0 are precisely the singular (= non-invertible) matrices in L\~,,(F). The same remark holds for any central simple A since AF = h f m ( F ) . Then the integer m in (3.11.1) coincides with the degree of A. It is well known that n ( z ) = det(&) is a prime polynomial in F [ e l 1 , . . . , (,,,:. . . (,,,I. It follows that n ( x ) is prime in F[<]for any central simple A. Hence (n (z))is a prime ideal in F[<].This implies that ( n ( x ) ) is the ideal of S.4 and S,l is a n irreducible projective variety for any central simple A. Moreover, the coordinate algebra F[S.4]= F[[]/(n,(z)) and the field of rational functions F(SA)O of SAis the subfield of the field of fract,ions F ( S A ) of F[SA] consistirig of t h e elements of F ( S A ) that are l~omogeneousof degree 0 (in the grading induced from the standard grading of F[E]). We call F(SA)0the (projective) n,orm field of A. We shall show that this field is a generic splitting field of A by relating it to another field that we shall define. IVe observe that tlie irre~lucibilit~y of n(x) in F[[]implies the irreducibility A]. It follows that na, (A) is irreducible in F([) [A] of rrLx (A) = n (A1 - x ) in F[<, arid lience F(()[x] is a subfield of AF(<).NOWwe have the following

.

Theorem 3.11.2 (Heuser [ 7 8 ] ) .F(<)[x]= F ( [ ,x ) is isomorphic to n simple transcendental extension of F ( S n ) . Proof. Let 1 = Ca,u,,a, E F. and put Theri F((1.. . . . <,,.z) = F ( y 1 , . . . , y,, r ) and

Since m , (n.) = 0. we have

3.11. Norm Hyperstirface of a Cerltral Simple Algebra

Hence t r deg F ( y l . . . . , y,)/F

5 7z - 1 and

t r deg F ( [ , x ) / F = tr deg F ( y l , . . . .y,,rc)/F 011the

/F

>

139

5 n.

(3.11.6)

other hand. since the 6, are algebraically independent. t r deg F(<. x) Hence (3.11 7) tr deg F ( E . x ) / F = n .

rl.

It now follows that tr deg F ( y 1 , . . . ; y,)/F = n - 1, and F(y1,. . . : yn x) is a simple transcendental extension of F ( y 1 , . . . , y,). NOWlet 71;. . . : 17, be indet,erminates and consider the homomorphism u of F [ q l , .. . , q & ] / F onto F [ y l , .. . . y,,]/F such that vi -, yz.1 i 5 n. By (3.11.5), ker u contains the prirrie ideal ( ~ , ~ (. v. . ~, v, n ) ) Hence we have the llomomorphism u' of F[q]/(r,,(q)) + F [ y l , . . . y,,] such that a (r,(v)) -, va; a E F[q]. Since tr deg F [yl , . . . y,]/F = n - 1 it follows that u' is an isomorphism (ZarisliiSamuel. v. I , Theorem 29 on p. 101). It follows that ker u = ( ~ ~ ( 7and )) hence

<

+

!

"

F [ l / l l ( ~ r n ( ~ )F) [ Y I ; .. . , yn]

" [a.

and F [ y l , . . . . y,] F Then F(<.x) = F ( y l , . . . , q,. x) is isomorphic t o a = F(SA). simple transc~ndentalextension of

~(c)

S3,'e recall that F ( S A ) is a simple transcendental extension of F ( S A ) ~ (Lemma 3.6.1). Hence F(<)[x]is rational of transcendency degree two over F ( S A ) ~Since . tr deg F ( < ) [ x ] / F= n = m,2;tr deg F ( S A ) ~= / Fn - 2. Since F(<)[x]is an m-dimensional subfield of AF(0 this is a splitting field for AF(<). Hence F(<)[z:] is a splitting field for A. Siilce F(<)[x]is rational over F(SA), it follows that F(SA)ois a splitting field for A. n(x) = det x. Now let A = Al,,(F). As before, we have z = C
of 0 onto 0'whose inverse is

Both p and p-l are regular. Hence \i7rhave

140

III. Galais Descent and Generic Splitting Fields

Theorem 3.11.10. SA for A == Mm(F) is birationally equivalent to the projective space PV where V is m 2 - 1 dimensional over F. Hence F(SA)O ~ F(PV)o is rational over F of transcendency degree m 2 - 2.

The arguments used for the Brauer-Severi variety VA can be repeated to establish the following facts: (i) F(VA)O is a regular extension of F, (ii) If F' is any extension field of F then F' (VApl)O == F' . F(VA)O. We leave it to the reader to check this. Also the proof of Theorem 3.8.7 carries over to prove Theorem 3.11.11. Let A be central simple over F, SA the norm hypersurface of A, F(SA)O its field of rational functions. Then: (1) F(SA)O is a generic splitting field for A, (2) F(SA)O is rational over F if and only if A ~ 1. (3) F(SA)O is a regular extension of F and if F' is any extension field of F then F'(SAFI)O == F' ·F(SA)O. (4) F' / F is a splitting field if and only it F' ·F(SA)O is rational over F'.

3.12. Variety of Rank One Elements Let V be an m-dimensional vector space over F, V* == Homp(V, F), A == End p V. We can identify A with V* (2)p V by using the standard isomorphism mapping u* (2)p V, u* E V*, v E V into the linear transformation x 'v'-7 (u*, x)v where (u*, v) == u*(v). Then the set p(A) of elements of rank one (e E A with [eV : F] == 1) is the set of u* (2)p v, u* =J 0, v =J o. We have (3.12.1) It follows that the left ideal A( u* (2) v ) == u* (2) V and the right ideal (u* (2) v )A == V* (2) v. These are m dimensional, are minimaI, and every minimalleft or right ideal has the form indicated. We have (V*

(2)

v)

n (u*

(2)

V)

== F(u*

(2)

v)

==

(V*

(2)

v)(u*

(2)

V)

(3.12.2)

which is one dimensional over F. We now give another characterization of the set p( A) of rank one elements. Lemma 3.12.3. p(A)

==

{e E A I e =J O,eAe == Fe}.

Proo/. By (3.12.1), (u*

(2) v)(x* (2) y)(u* (2) v) E F(u* (2) v). Conversely, let L~ (2) vi E p(A). We may assume the are linearly independent and the vi are linearly independent. Then r ::; mand there exist Ui E V, vi E V* such that (u;, Uj) == 8ij == (vi, Vj). Then

e ==

u;

(Eu;

This implies that r

u;

(2)

Vi)(v'k

(2)

uk)(Eu;

== 1. Hence p(A) ==

{u*

(2)

Vi)

== uk

(2)

v I u*

(2)

vk.

i= 0, v i= O}.

o

3.12. Variety of Rank One Elements

141

In any associative algebra A we define

so U,z = a r a . Also put Ua.6= LTa+b- Ua - Ub SO Ua bx = azb

+ bza and

\Ye define a E A to be of rank 1 if a # 0 and U a A = Fa. Then if A = EndFV or hI,,,(F),a E A is of rank one in the sense just defined if and only if a is of rank one in the usual sense. Now let ( u l , . . . . u,) be a base for A / F and let

F . Then for the generic element x = C<,u, A r ( < )we have

* / 2 k i rSZ3ktE

(E,

indeterrninates) in

where

Pllt f j k e ( 6 ) = Ej
Then it is clear t,hat a = Gaia, E in AE if and only if a # 0 and

AE for an extension field E / F

(3.12.9) is of rank 1

for all J . X. 1. Since the polynomials f j k i ( E ) are homogeneous they can be used to define a projective F-variety RA as the set of points p = Fa in P A F , F an algebraically closed field containing F such that f J k a ( a )= 0 for all J . k , !. If E is any extension field of F then the E-rational points on RA are precisely the p = E a sudi that a is of rank 1. The connection between the geometry of RA and the algebra A is given b y the follou~ing

Theorem 3.12.11. The central simple algebra A contains elements of rank 1 if and only i f A N 1.

-

"

Proof. If A 1 so A = EndFV V* @ F V then A contains elements of rank 1, u* E u . Conversely, assume A contains an element a of rank 1. The

142

111. Galois Descent and Generic Splitting Fields

condition &A = Fa # 0 carries over to U,Ap = Fa # 0 for the algebraic closure F of F . Thus a is of rank 1 in AF and hence [aAF : F ] = nb. It follows tliat ( L Ais ail m-dimensional right ideal in A and we have seen (p. 111) that the existence of such a right ideal implies A -- 1.

A n immediate consequence of this result is that RAcontains an E-rat,ional point for E an extension field of F if and only if E is a splitting field for A. Now let 11' and W be vect,or spaces over F of respect,isre diinensioiis 771 and n. Then [V8~W : F ] = m n and me have the subset P V x P W of P ( V BW ~ ) Fconsisting of the points F ( v % w ) , O # v E V = V p ;0 # w E TVF. If ( e l : .. . . e m ) and ( f l , .. . , f,) are bases for V and W then (ei f,) ordered lexicographically is a base for V @ 2'7. It is a well-known fact from linear algcbra that F ( C n t k e ,8 f k ) , a,k E F : is coritairled in P V x PLV if and orily if

-

Then P V x P W is a projectjive variety defined by the set of homogeneous polynomials CLkCje - CjkC2! E F[<] F [ < I .~. . : < i k > .. . , Cmn]. Let Z' be the ideal in F[C] generated by the polynomials Cik& - CjkCLt.Then nilrad Z' = Z = Z ( P V x P W ) . WTe have a honlomorphism 8 of F[C] into F [ [ :q] FIE1:.. . , E m , 711,.. . , q,l]: Ei. Y ~ / C indeterminates, such that Q ( C i k ) = Etqk The irnage of 0 is F[Eq]= F[[1?11,. . . , E7qlc; . . . Emqn] wliicli is a domain. Hence ker Q is priinc in F[C] and so ker 0 = nilrad ker 0. It is clear that Z' c ker 0. Hence Z = nilrad Z' C nilrad ker 0 = ker 0. On the other hand, if f ( C 1 1 , . . . ; Cik. . . . : C,,) E ker 0 then f ([iq, : . . . , Ez'rlk. . . . . Emqn) = 0. Hence f ( a l b l ,. . . , nibk. . . . : a, b,) = 0 for all a, and bk. Thus f E Z and ker 0 c Z . Hence Z = ker 0. Hence the coordinate algebra F [ P V x P W ] = F [ C ] / ZE F[
--

+

+

Theorem 3.12.13. If A is cent~,alsimple of degree m then RA is irreducible and t r deg F ( R A ) ~ / = F 2(m - 1 ) .

3.12. Variety of Rank One Elements

143

Proof. Let E be a finite dimerlsional Galois splitting field for A. Then AE = Afv,(E) and we can conclude the irreducibility of RA from that of RA, (p. 3.38). Also the proof of Lemma 3.7.7 carries over to show that E(R*,)o = E Z r F ( R A ) .Then t r deg F ( R 4 ) o I F = t r deg E ( R A , ) / E = 2 ( m - 1). \fTe Carl now state the analogue of Theorem 3.8.7 for RA.

T h e o r e m 3.12.14. Let A be central simple over F , RA its variety of elem,en,ts of rank 1. F ( R A j o its field of rational functions. Then: (1) F ( R n j o ,is a generic splitting field for A. (2) F ( R A j o is rational over F if and only if A 1. (3) F ( R A ) o is a regular extension of F and if F' is any extension field of F then F ' ( R A , , ) o = F' . F ( R A ) . (4) F 1 / F is a splitting field for A if and only if F' . F ( R A ) is rational over F'.

-

We ornit the proof which is similar to that of 3.8.7. In the remainder of this section we investigate relations connecting the varieties V A ,SA and RA arid their fields of rational functions. lve shall require the irnportant concept of the product of projective varieties. Let V be a subvariety of P V with ideal Z ( V ) , W a subvariety of P W with ideal Z ( W ) and let V x W denote the subset of P V x P W of points F ( v @ w),FV E V . Fu:E W . We claim that V x W is a projective variety defined by the following set of polynomials in F [
Let K denote the ideal in F [ ( ] generated by these polynomials and suppose h ( p ) = 0 for all h E K . Sirice K > Z ( P V x P W ) it follows that p = F ( C a , b k e , @ = D for f k ) for a,, bk E F . I\Ioreover, f ( a l b k . . . . , a m b k ) = f ( a l . .. . , a,)bpg all f E Z ( V ) and 1 5 k 5 n. Since some bk # O we have f ( a l . . . . , a,) = 0 for all f E Z ( V ) and similarly g(b1,. . . b,) = 0 for all g E Z ( W ) .Then p E V x W . Conversely. if p E V x W then h ( p ) = 0 for h E K. Hence V x W is a projective F-variety and Z(V x W ) = nilrad K . W e car1 iiow state the following

.

T h e o r e m 3.12.16 ( P e t e r s s o n [84]). (1) If A is central simple t h e n RA is isomorphic to V A x VAo (VA. the Braum--Sever2 v a ~ i e t yof A). (2) F ( R A ) is ~ r ~ t i o n ~ aouer l F ( V A ) of ~ transcendency degree m - 1. Proof. ( 1 ) \T7e note first that since the right ideals of A$ are the left ideals of A , = l\.f,(F) we rnay regard VAo as the subvariety of Gr,(A) corresponding t,o the minimal left ideals of AF. The results at the beginning of the section shorn t,hat if F a E RA then this point determines the minimal left ideal AFn and the minimal right ideal a A F of AF> hence, a point of V A x V,40. Also if R arid L are respectively irliiiimal right and minimal left ideals of AT then R n L = RL = Fa E RA. We obtain in this way a bijection of RA onto

144

111. Galois Descent and Generic Splitting Fields

VA x V.4~.Explicitly, if ( w l , . . . , w , ) base for AFa then

is a base for aAF and (wi, . . . , ulh) is a

is bijective of RA ont)o Vil x VAo with inverse

We shall show that cp and pP1 are regular which will prove that RA and VA x VAOare isomorphic. Let ( u l . . . . . un)be a base for A I F , x = C: [,u,, [, indeterminates. and let 1 5 il < . . . < i, 5 n. Then

where (i) = ( i ~ : .. . 'i,), ( j ) = ( j l , . . . , j,),l I jl < . . . < j, 5 n , and 17(,).(3) E F [ [ ] is homogeneous of degree m in the ['s. Similarly, xu,, A . . . A xuim = ((l).(j)ujlA. . .Aujm where <(i),(,) E F[[]is homogeneous of degree m. Then {a E AF u i , a A . . . A u , _ a # 0) and {a E AF / auil A...Aaui, # 0) are Zariski open. Moreover, if a # 0 in AF t,hen the left ideal AFa and right ideal aAF have dimension 2 m so there exist 21;. . . , i, such tha,t u,,a A , . . A 7l3,_a# 0 and jl: . . . ,j , such that auj, A . .Aau3_ # 0. Hence we have a n open cover of A, consisting of the open sets O(i),(3) = {a E AF / au,, . .Aa,uim# 0 , v,,a A . . . A uj,n # 0). Let a E RAn O(i),(j).Then the map

x(j)

of P A into Gr(A, m ) x Gr(A, m ) is clearly regular. Since 9 is the restriction of this map t o R A ,p is regular. To complete the proof of (1) we need to show that p-l is regular. This will follow from the following more general result. Lemma 3.12.21 (Saltman). Let V be an n-dimensional vector space over F, U the subset qf Gr(V, m ) x Gr(V, rn), 0 < m < n of points F ( w l A . . .AUI,) 8 F(w: A . . . A w;) such that the intersection CFW, n CFW;of the corresponding subspaces of V is one dimensional. Then U is an open subset of an F-variety and the map $ : F ( w A~ . . . A wTn)@ F(w:A . . . A w k ) (3.12.22) --t Fuji n c F w : of U into P V ,is regular defined over F (as for varieties). Proof. Let ( u l , . . . , u,) be a base for V I F ,

3.12. Variety of Rank One Elements

145

the corresponding bases for Am(V) and An'+' (V). Ally u r E Am(VF) defines a linear map ~ ( w :) v --i v A w of VF into An*l(VF). Using the foregoing bases, r ( w ) can be represented by an s times n , s =

(

.

l ) matrix ~ ( u , )

n-hose cntries are homogeneous linear combinations w'ith coekicients in F of the Pliicker coordinates p,, ,_of u l = C,l, <,m p,, ,_uL1A . . . A uz,, . Let iu' he a second element of iln(VF). Then we have the 2s by m matrix

(1

and it is clear that ker y(w) n ker y(w') is the set of vectors that T(w, w')

=O.

C: a,u,

such

(3.12.25)

an

Moreover, [(CFw, n 'IFwl) : F] = n

-

rank r ( w , w').

(3.12.26)

Since the entries of T ( w , w') are homogeneous linear combinations with coefficients in F of t,he Pliicker coordinates of w and UI' we obtain a subvariety W of Gr(V, n ) x Gr(V, n ) by requiring that the rank of ~ ( u w') I , < n and an open subset U of W by requiring that the rank is n - 1. By (3.12.25), W is the #0 set of points ~ ( 8ww') of Gr(V, n) x Gr(V,n) such that C F Wn~CFW; and U is the subset such that C F w i n CFwi is one dimensional. Then we have the map $ of U into P V defined by (3.12.22) and we have seen that q5 : F ( w @ lu') E U -, Fv where 71 = Caiu, is as in (3.12.25). Now let 1 5 jl < . . . < jnP15 2s and let Uj1...37L-1 be the subset of U defined by the . . condition that there exists an ( n - 1)-rowed minor A;!:,'"' # 0 in r ( v , v'). Thus Uj, ,.,. ,jn-, may be vacuous hut .in any case this is "an open subset of U . and U = UU31,,,3n. Let qi = (-1)'A;t::i3:;l. Then Caoqj = 0, 1 5 i 5 2s, since 0921

" " "

'3171

"""

aj271

......

a3n-ll . . - ail

" " "

= 0.

(3.12.27)

ajn-~n

ain

This is clear if 2 is one of the jk and otherwise it follows since every n-rowed ininor of r(w,w') = 0. Since the (n - 1)-rowed minors of a matrix are homogeneous polynornials with coefficients in F of the entries of the matrix it follows that w is regular. This lemma completes the proof of (1) of Theorem 3.12.16 since the Brauer-Severi varieties VA and VAo are subvarieties of Gr(A, m) and p-l =

1 VA x VAo.

146

111. Galois Descent and Generic Splitting Fields

F ( V A ) .~F(VAo)o. Since F ( V A , ) ~is a splitting (2) By (11, F ( R A ) o ~ Thefield for A'. hence for A. F(VA)O. F ( V A ~is) ~rational over F ( V A ) by orem 3.12.13 (4). By syi-nmetrv. F ( V A ) .~F(VAo)0 is rational over F(VA)0. Hence F ( R A ) is~ rational over F(VA)O.Since t r deg F ( V A ) ~ / = F i n - 1 and t r deg F ( R A ) 0 / F = 2(m - I ) , t r deg F ( R A ) O / F ( V A ) O = m - 1. We shall obtain next a relation between the norm hypersurface SAof A and R A . This is given in the following Theorem 3.12.28 (Petersson, loc. cit). If A Is centml simple of degree m, fhen SAis birationally equivalent to P W x R A luhere I+' is an (nx - 1)'dimensional ~iectorspace over F .

We require several lemmas for the proof. The first of these is classical (see e.g. IVedderburn's [34]. p. 67). Lemma 3.12.29. If n(!) = det L = 0 for !E A = El1dFV then the reduced adjoint L# (p. 26) has rank 5 1.

Proof (Craig Huneke). IVe recall that if L has the matrix L = (Lt3) relative to the base (el, . . . , em) for V then the (i, j)-entry of the rnatrix L# of L# is (-1)i+jL3, where Lgi is the minor of Lj, in L (BAI, p. 96). Hence the matrix form of the lemma is that of det L = 0 then rk L# 5 1. Now let X = (Ei3), a n indeterminate, and consider the algebra F[<]= F[<]/(det X) where F[E]= F [ < l l , .. . , [ij, . . . ,Emrn] and = Ez3 (det X ) . (Note t>hat F[(] is just the coordinate algebra of SAfor A = hl,(F).) If det L = 0 for L = (el,) E M m ( F ) then we have the specialization (homomorphism) v~ of F [ ~ ] /onto F F such that !ij,l 5 vl, 5 m,:Since t,here exist L such that L1l # 0, the minor ~ 1 of1 Ell in X = is # 0. Hence rk x = m - 1. Now X X # ( d e t X ) 1 (BAI, loc. cit.). Hence XX# = 0. We have seen in Section 10 that (det X ) is a prime ideal. Hence we have the field F(<) and x E ,%fm(F(()) has rank m - 1. Hence its null space is one dimensional. Since the equation X X # = 0 implies that the collimns of X# are in this null space, rk X # 5 1. Then rk L # 5 1 by specialization.

cis

ti,

+

(c,]

If L = u* g t l , u* # 0.7' # 0. as above, then by (3.12.2). e2 = (u*,u)T and, as is well-known. (w*, v) is the trace t(!). Hence ~f rk != 1 then l2 = 0 if

t(P) = 0 and e = t(l)-'! is idemgotent if t ( l )# 0. If r is a n idempotent in a n algebra A we have the two-sided Peirce decornposition of A relative to e.

pAe = U,A is a n algebra with e as unit and (1 - e ) A ( l - e) = U1-,A is a n algebra with unit 1 - e. Also eA(1- e) (1 - e)Ae = Ue 1-,A. If A = hfr,(F) and rk e = 1 then there is a matrix base (e,,, 1 5 2 . 3 < m) with e l l = e. Then UeA = F e l l . Ue,l-eA = C z > l F e l t + C z > l Fez1 and Ul-eA = Cz,>I FezJ.

+

3.12. Variety of Rank One Elements

147

Hence U l - , A r lZ/I,,-l(E). If b E Ul-,A we denote its reduced norm and reduced adjoint in U1-,A by n l ( b ) and b#l. Then we have

Lemma 3.12.31. If cr E F and b t Ul-,A then n,(cre ( a e + b)# = n l ( b ) e ab#'.

+

+ b) = crnl(b) and

Proof. The first is clear. To prove the second it suffices (using the Zariski topology in A F ) to prove the formula in the case in which n ( a e b) # 0 . We have

+

Since n ( a e + b ) # 0;ae+b is invertible. Then (3.12.25) implies that (ae+b)# = n,l(b)e + ab#'. We require also

Lemma 3.12.33. If b E A = A/l,,(F) satisfies

th,en e = t(b#)-lb# is a 7.0,7bk one idempotent; b E Ul-,A; n l ( b ) = t ( b # ) so b Is znnertlble i n U1-,A. Proof. B y 3.12.25, n ( b ) = 0 implies rk b# 5 1. Since (b#)' = t(b#)b# # 0 ;rk b# = 1 and e = t(b#)-lb# is a rank 1 idempotert. Since bb# = n,(b)l = b#b? Ul-,b=

(1 -e)b(l - e ) = b - e b -

= b - t(b")-lb#b

-

be+ebe

+

t ( b # ) ~ l b b # t ( b # y 2 b # b b # = b.

Hence b E U1-,A. By 3.12.31, b# = n l ( b ) e and since b# = t ( b # ) e , n l ( b ) = t ( b # ) # 0 . Then b is invertible in Ul-,A. TVe can now give the

Proof of Theorem 3.12.28. If u has rank 1 in A = A F then e = e ( u ) = t ( u ) - l u is the unique idempotent in Fu and we have the Peirce decomposition (3.12.30) for A relative t o e . Tlien ker U I - , = U , ~ ? + U , , ~ - , A so [ker U I - , : F] = 2 m - 1 . TVe now choose a rank 1 idcmpotent eo E A and let It' be a subspace of A I F such that W = W> is a complerrient of ker u,-,,A in A. Then [W : F] = (nz - 1 )' . Consider the subset X of R q defined by X = { F u E RA I t ( u ) # 0 , ker U1_,(,) n W = 0 ) .

(3.12.35)

Evidently F e t X and it is readily seen that X is an open subset of RA.If F I I E X then [u~-,(,)A: F] = ( m- 1 ) 2 and U1-,(,) maps I@ bijectively onto C;-,I,lA. Next let 0 be the inverse image of X under the adjoint map:

148

111. Galois Descent and Generic Splitting Fields

0 = { F G E S A / t(.o#)# 0. ker U1-,(,#)

n W = 0).

(3.12.36)

Then ~ ( - 1e) E 0 and 0 is an open subset of S A . We have a map cp of X x PW such that cp(Fu @ F w ) = FU where 21 = Ul-,(,)w. B y (3.12.30). n ( v ) = 0 and b y the secoild condition on u in (3.12.34), v # 0. Hence p maps X x PW into S A . Finally, let 0' = cp-l(O). We have seen that if F I L E X then Ul-,(,) maps W bijectively onto u ~ - , ( ~ AHence . there exist w E W so that v = Ul-,(,)u1 is invertible in Ul_,(,)A (e.g. v = 1 - e ( u ) ) . Then, by 3.12.30. tl# = nl ( v ) e ( u )and nl (21) # 0 since 11 is invertible in Ul_,(,,)2. Hence. by 3.12.32, n l ( v ) = t(v-#) and e ( v # ) = r ( u ) . Then ker Ul_,(,+t) n W = 0 so F v E 0 . Since p ( F u @ ~ u = l ) F v , v = Ul-,( ~ U J F, u g F w E 0' and 0' # cp. It is readily seen that cp is a regular map of X x PW. Hence 0' = cp-l(0) is open in X x PW and hence in RA x PW. We now have the map cp of 0' (9/ 0')onto 0 sucll that ,J,,

Kext let F v E 0 . Then, by 3.12.33. v E U1-,(,)A. Moreover, U1-,(,g) bijection of W onto Ul-,(,#)2 so we have the inverse map (Ul-,(,#) of U1-,(, + ] Aonto W . Hence we have the map 7+0 of 0 such that

I W is a Iw)~'

It is readily seen that ?1. is regular. Since

11..(~v) E 0' and cp$ = l o . Next we claim that $cp = l o t . Let F u 8 F w E 0' Then cp(Fu @ F u . ) = F U ~ - , ( , ) U ~ and q!!cp(Fu.F w ) = Q ( F U ~ _ , ( , ~ W ) - = Fu' X Fw' whcre u' = (Ul-e(,)w)t = nl(Ul--e(,jw)e(u). Hence Fu' = F u . Then Fur' = F(Ul-,(,,) / W ) - l U l - e ( , , ) w = Fu' so t5ip = l o ) .Hence cp is regular s and is regular. Then S A and P1I7 x RA are birationally and $ is ~ t inverse equivalent. Now the field of ratioilal functions on P W x RA is isomorphic to F ( [ ) o . F ( R A ) where ~ F ( < )is rational over F of transcendency degree m2 - 2 m over F . Since this field is rational over F ( R A ) of ~ transceildency degree m2 - 2m we can conclude the following consequence of Theorem 3.12.16. Corollary 3.12.42. F ( S A ) is ~ isomorphic to an eztension of F ( R A ) o that is rational of transcendency degree m2 - 2 m over F ( R A ) .

If we combine this with 3.12.16 ( 2 ) that F ( R A ) is~ rational over F ( V A ) o of transcendency degree m - 1 we obtain Corollary 3.12.43 (Saltman 1801). F ( S A ) ~is isomorphic to a rational extension of F ( V A ) of ~ transcendency degree m2 - m - 1 over F ( V A ) o .

3.13. The Brauer Functor. Corestriction of Algebras

149

3.13. The Brauer Functor. Corestrict ion of Algebras Let A be an algebra over a field F, a a homomorphism of F into a second field K . We may regard K as an F-module in which

Regarding K as F-module in this way and A as F-module as given, we can define the tensor product

which is an algebra over F (since both K and A are). Now A O ,can ~ also be regarded as an algebra over K in which k(! $3, F a ) = k! @,,F a. The special case we have been considering heretofore is that in which K is an extension field of F and a = L the injection of F into K . It is illuminating to see what A o , means ~ concretely in terms of bases. Let x l . . . . , x, be a base for the algebra A / F with multiplicatio~ltable

Then it is readily seen that A,,K has the base (1 $3 zi wit11 m~iltiplicat,ion table

/ 1
n) over K

Thus the passage from A to A,,K is realized by applying a to the "constants of ~nultiplication"- , , k and replacing the field F by the field K . Either by arguing with bases or by abstract considerations we see that if L is a third field and r is a homomorphism of K into L then r r is a homomorphism of F into L and we have the functorial property ( A o , ~ ) r2, ~ A,, under a map such that l c 3 (k ~ 8~a ) --i (rk)k 8~a , k E K , l E L, a E A. If B is a second algebra over F then (A @F B),,K A u , %~K B,,l{. Since lldm(F) has a base with multiplication constants in the prime field it is clear that Ad,,(F),,I( E hl,(K) and since central simplicity of an algebra A can be characterized by the condition A %F A0 2 M m ( F ) for [A : F] = m, it follows that if A is central simple over F tllerl A@, K is central simple over K . It follows that the map [Ao,~l

(3.13.5)

is a homomorphism of Br(F) into Br(K). This defines a functor, the Brauer group functor, from the category of fields to the category of abelian groups. Thus far we have considered "extension" of the base field of an algebra. We shall define next an important process, the corestriction map, that associates with an algebra A over a field K that is finite dimensional separable over F an algebra over F. corKIFA.

150

111. Galois Descent and Generic Splitting Fields

Let K be a finite dimensional separable field over F . E a finite dimensional extension of I< that is Galois over F If [ K : F] = n we have n homomorphisms of K / F into E / F . Let 111 be the set of these and let G = Gal E / F . If a E 121 and a E G then aoi E 1LI. Since any cu can be extended to a n automorphism of E l F . 121 can be identified with the set G / H of left cosets of G with respect to H = Gal E I K . Let A be an algebra over K.For any a E M we define the algebra A, E gnK A over E . If a E G me have a map a ( a ) of A, into A,, such that ~ ( ~ @ ) ,( I<e a ) = a e g o a ,a.~I11 terms of bases, let ( 2 , ) be a base for A / K so ( 1 & r , ) is a base for A,/E. IVe now write z, for 1 @ z , so the elements of A, have the form C e , z , , e, E E . Then a ( " ) ( ~ e , z , = ) C ( a e , ) x , . The map a ( a ) is a n isomorpliisrri of algebras over K arid is a a-sernilinear trarisforrnation of A,/E onto A,,/E. I i e now form the E-algebra

-

A

= A,,

@E

A,, B E

"'

@ E Aan

where M = {cwl,. . . , a , ) . Slre require the

Lemma 3.13.6. Let v = Vl XE . . . BE Vn where V , is a vector space over E . Let s E S, and o E G . Suppose for each i, 1 < i 71: we have a a-semilinear transform,ation, n, of V , into V,(,). Th,en there is a ~ ~ n i q ua-semilinear e transformation 6 of v into itself such that

<

Proof. For e E E we have

Hence by the universal property of Vl @ E . . . %E Vn we have a n additive group endomorphism 8 of v such that (3.13.7) holds. By (3.13.8). this is n semilinear. LIoreover, the uniqueness of i is clear. IVc apply this t o A where s is the permutation such that ( g a l . . . . , g o n )= Then i is a n autonlorphism of A as algebra over F and . ... 67 = iT? for T E G Then if B = Irlv G for G = (6 1 a E G). B is a n algebra over F such that A = EB 2 BE. Thus B is a n F-form of A. The algebra B is independent of the choice of the Galois extension E/F used to define it. It suffices to show independence if E is replaced by a Galois E f / F containing E . Then the monomorphisms of K / F into E1/F map K / F into E/F so we can identify A f with the set of monomorphisms of K / F into E 1 / F . I4'e have AEr(ej 2 Ef @E AE(aj (by e' 8~ (e 8~a ) --, ele C j i a~ ) and

3.13. T h e Brauer Functor. Corestriction of Algebras

151

Thus we may identify A with an E-subalgebra of A' such that E'A = A' and E' arid A are linearly disjoint over E . Also if a' E G' = Gal E'IF and 8' is the correspondi~lga-sernilinear aut,oinorphism of A' then 8' I il = 8 for some a E G and every 6,a E G , can be obtained in this way. It follows that B c B' = Inv G',G' = ( 8 ' ) . Since E B = A and E I A = A' we have E'B = A'. Since E'B' E'@ F B' arid B c B' it follows that B = B'. We shall now call B,which is independent of E, the corestriction c0rICIFA of A I K . We have [corKIFA : F] = [ A : El = [AE : Eln = [A : KIn. It is readily seen that the map A --i corKIFA is functoria,l from the category of K-algebras to t h e category of F-algebras. In many ways cor behaves like the norm for finite dilnensional field extensions, namely, we have the following properties whose proofs we leave t o the reader.

"

C O ~ K / F A'@F I C~I.K/FAZ (i) COI.I<~F(AI @ K A2) (ii) If L is a subfield of K then,

The foregoing definition simplifies somewhat if K it,self is Galois over F. I n this case we may take E = K . Then M = G = Gal KIF. The case of importance for the study of involutions (see Section 5.2) is that in which K is a separable quadratic extension of F . Here if ( e l , . . . , e m ) is a base for AIK t>hen (ei ;O e j 1 < i , j 5 rn) is a base for A and if : a --, li is the automorphism # 1 of K then 8 is Ca,,,(e, @ e j ) --, Cai,(s, 8 e , ) . Then corKIFA is the Fsubalgebra of A consisting of the elenlents CatjeiX e, in which a,, = aij. We now consider t,he following

Ezample. Let K be a quadratic extension of F. char F # 2 so K / F is separable and. as above, let CT be the automorphism # 1 of K I F . Let B be the algebra of dual numbers over F , so B = F1 8 Ft where t2 = 0 and let ,4 = B K . The algebra A" is A with the K-module action given by k ( a + bt) = ka ibt. In A we denote t 8 1 by t and 1 8 t by t'. Then A has the base (1, t , t', tt' = t't) with t2 = 0 = t f 2 .Then corKIFA is the F-subalgebra of A consisting of the elements a + bt + bf' + Btt' where a , p E F,b E K. Using the norm analogy orie might expect corKlrA E B XF B. This is not the case. For. the set of elements z E cOrKIFA slicli that z2 = 0 is the 1-dimensional subspace F(ttl). On the other hand. the subset of z E B @F B such that z2 = 0 is {pt 6tt') u {rt' stt'), 0.7.6 E F . Since this is not a subspace of B 8~B it follows that B 8 3 B corKIFA.

+

+

+

152

111. Galois Descent and Generic Splitting Fields

We shall be interested primarily in the case in which A is central simple over K. Then AE = E SKA is central simple over E and every A&(-)is central simple over E. Hence A is central simple over E and corKIFA is central simple over F. We shall show that we have a well-defined homomorphism [A] --, or^,^] of B r ( K ) into Br(F). For this we follow an exposition given in some unpublished notes of Tamagawa's to whom we are indebted also for the foregoing definition of the corestriction of an algebra as well as the above example. A more general definition of corestriction is given by Carl Riehm in [70].See also J. Tits [71], p. 208. Aga,in: let A be simple with center a finite dimensional separable extension K I F . We choose a splitting field E I K for A that is finite dimensional Galois over K . By extending E further if necessary we may assume E is Galois over F. Let G = Gal EIF, H = Gal E I K and let M be the set of monomorphisms of K I F into EIF? which we saw can be identified with the set G I H of left, cosets OH of H in G. We choose a set of representatives of these cosets, or equivalently, for each a E 1Z.I we choose a, E G such that a, I K = a. Then for CT E G:a: E M we have

where X,(a) is a uniquely determined element of H . Thus each a E M defines a map a --i X,(a) of G into H (depending on the choice of the set of repre~ , (a). sentatives). For T, a E G , we have TOO, = T , , ~ ~ X , (TO)= o ~ , ~ X (r)XO( Hence (3.13.11) Xa (70) = Xun (7)Xa ( a ) Since AE 1 we can identify AE with the algebra EndEV of lincar transformations in an m dimensional vector space VIE. For each X E H we have a A-semilinear transformation sx of VIE such that

and A is the K-subalgebra of EndEV defined by the condition s x a s x l = a. X E H (see p. 3 6). IVc now take n = / A f copies V,/E of VIE, a E M , and we assume that for each cu E M we have a bijective a,-semilinear map .c --i z, of V onto V,. Then for a E G and a E M we define a map up) of Ve into V, by

By (3.13.10) this is a-semilinear. Hence, by Lemma 3.13.6, we have a Osernilinear transformation of V = Val @ E . . . @E Van into itself such that for x, E V ,

where ytn, = T

L ~ - ~ ~ ~ ) Z ~ , ~ - ~ , ,

(3.13.15)

3.13. The Brauer Functor. Corestriction of Algebras

153

By using (3.13.10) and (3.13.11) we can calculate:

This implies that iiTu,, = CT ,,fir, where =

IT

~ T O ~ ( & Q ( A,T(.))) .

(3.13.17) (3.13.18)

n t A1

If a E EndEV then we define a, E End V,.a E hf, by a,x, = (ax),. If a E A C EndEV, as^ = sxa,A E H . Then by (3.13.13). u?)a,x, = a,, U ( ~ ) It Z~ follows . that if a, E A,, 1 z. n , then

< <

where a,, = a-la,. On the other hand: by (3.13.7): K(al,, @A. . . Cjr a n o n ) = al,,, @ . . . 8 Since the elements al,, 8 . . . @ anan span A over E and c? and the map z --t fiiL,zfi;l in A are both a-semilinear it follows that 6coincides with z --t ii,ziiil. This implies that the algebra of linear transformations in VIE that commute with the 6, coincides with cOrKIFA. A consequence of the foregoing result is that if A is split then cOrKIFA is split,. For, [A] = [ ( E / K ,H,!)lp1 so [A] = 1 + [C] = 1. In this case we may assume = 1 for A, p E H . Then i,,, = 1 and [ ( E I F ;G. !)] = 1 so [cOrKIFA] = 1. This result and the multiplicative property (i) implies that we have a homomorphism [A] --t [cOrKIFA]of Br(K) into Br(F). The result we have proved is that this homomorphism corresponds to the "transfer" homomorphism of 2-cohomology groups (Brown [82]. p. 80). We can state this result in t,he following explicit form:

Theorem 3.13.20. Let K / F be finite dimensional separable, E/F finite dimensional Galois containing K , G = Gal EIF, H = Gal E / K , M = G / H tlre set of left cosets of G relative to H. Let [&I E H 2 ( H ,E * ) and let [A] be the corresponding element of B r ( E / K ) . T h e n the element of H 2 ( G ;E*) corresponding t o [corKjFA] is [i]where 2 is given by ,9.13.18.

IV. p-Algebras

A p-algebra is a central simple algebra A over a field of characteristic p > 0 such that [A]E Br,(F). the p-th conlponent of B r ( F ) , that is, the exponent of A is a power of p (equivalently, the index is a power of p). The structure theory of these algebras was developed by Albert in the thirties and was presented in an improved form in Chapter VII of his Structure of Algebras (see also Teichmuller [ 3 6 ] ) .The culminating result of Albert's theory is that. in his terminology, any p-algebra is cyclically representable, that is. is similar to a cyclic algebra. This is proved in two stages: the first, in which it is shown that any p-algebra is similar to a tensor product of cyclic algebras and the second. in which it is proved that the tensor product of two cyclic p-algebras is cyclic. The first of these results is proved in Section 4.2. Purely inseparable splitting fields play an important role in the theory. We recall that K I F is purely inseparable if K I F is algebraic and its subfield of separable elements coincides with F, or equivalently, for any u E K there exists an e 2 0 such that ape E F . The exponent of K I F is e < cc if there exists an e such that ape E F for all a E K and e is minimal for this property. If [K : F ] = n < M then K has ~xponente such that pe 5 n. (See Section 8.7 of BA I1 for these results.) We shall consider purely inseparable splitting fields of p-algebras in Section 4.1. The main result here is that any palgebra has finite dimensional purely inseparable splitting fields and if e is the minimal exponent for such fields then the exponent of A is pe. This is proved by Albert at the end of his discussion. Here, following a n exposition in Saltman's thesis [76],it is proved at the beginning, in Section 4.1, by using the Frobenius map in Br(F) for F of characteristic p. If F is a field and G is a finite group then we say that G appears over F if there exists a Galois extension field E / F with Gal EIF G. If A is central simple over F then we say that Gappears over F in A if A contains a maximal subfield EIF that is Galois with G a l E I F G. For any p-group G, VCTitthas given in [36] a necessary and sufficient condition that G of order pf appears over F of characteristic p: If the dimensionality of F / P ( F ) over the prime field P is N, where P ( F ) = {P(a)= a, - a: / a: E F) and the minimum number of generators of G is f then G appears over F if and only if f 5 N. Saltman has proved in [77] that if /GI = pe and G appears over F of characteristic p then G appears over F in any cyclic algebra of degree pe. We shall prove the

"

N. Jacobson, Finite-Dimensional Division Algebras over Fields , © Springer-Verlag Berlin Heidelberg 1996, Corrected 2nd printing 2010

"

4.1 The Frobenius Slap and Purely Inseparable Splitting Fields

155

part of Witt's theorem that we require in Section 4.3 and Saltman's theorem in Section 4.4. Saltman makes use in his paper of a criterion for cyclicity due to Albert,. However, as we shall show in Section 4.4, Saltman's proof of his theorem actually gives a proof of Albert's theorem. These results are used in Section 4.5 to prove the teiisor product theorem for cyclic p-algebras. A question left open in Albert's theory was: Is every division p-algebra cyclic? A counter example to this has been given by Amitsur and Saltman in [78].This is based on a construction of generic abelian crossed product,^ that we shall discuss in Section 4.6. This is of considerable independent interest. In particular; we shall need it again in Section 5.7. The construction of non-cyclic p-algcbras will be given in Section 4.7.

4.1. The Frobenius Map and Purely Inseparable Splitting Fields The Frobenius map which we shall now define is a special case of a tensor product which we considered in Section 3.13. Let A be an algebra over a field F, a a homomorphism of F onto a second field K . Then we can form the F-algebra.

where K is regarded as F-module by defining a k = (acr)k. for a E F, k 6 K (see Section 3.13). We can regard A,,Ic as an algebra over K and we obtain a llornornorphism of B r ( F ) int>oBr(K) by mapping [A]--, [A,.K]for A central simple over F . We now consider the special case of this in which K = F is a field of characteristic p and a is t,he Frobenius endomorphism Frob : a -, ap. The corresponding algebra defined by (4.1.1) is A(") = F @ F ~ ~ A ~ . and F we have the Frobenills endomorphism [A] -, [A(")]of Br(F). A key result on this endomorphisrn is Theorem 4.1.2. Tlze Froben,ius endomorph,ism of Br(F) coincides with the p-po~uermap [A] -, [ A l p .

Proof. It suffices to show that if A = ( E , G, k) where E I F is Galois with G a l E I F = G then ( E , G, k)(p) E ( E , G; kp) since [(E;G, kp)] = [(E,G, k)]p. Let ( a l , . . . , a,) be a base for E I F and let G = {al,. . . , a,). Then the elements aiuD2,1 i , j 5 n, form a base for ( E , G, k) over F if (u,,, . . . , uOn)is a usual base for ( E , G, k) over E determined by G. Suppose a,aj = C, yijeae, uiaj = CeSijeae, a, kO2,, = CcE j j ~ ~ t awhere e "iz,g, 6i,t,~jjj,p6 F . Then the multiplication table for the base {u,u,,) is

<

156

IV. p-Algebras

For ( E , G. k ) ( " ) we have the base 1 E a,uuj , 1 5 table (1 @ a,umj) ( I E atlumj, 1 =

2, J

5 n with multiplication (4.1.4)

):( 6 3 z ' ~ ~ j a ~ ~ m ~ z %, naquo,o,, q ) P ( 1 ). e,m,q

Next consider ( E , G, kp). Since E I F is separable the elements a: constitute y&pa;. Also a base for E I F and we have the multiplication table ara; = aza7 = EL6:7ea; and aY(kmJ,,j,)P = CL(~,Jl,r)Pa:.It follows that if we use 3 (E.G, kP) we obtain the same multiplication table (4.1.3) the base a ~ u mfor Hence (E,G, k)(p) N (E,Gl k p ) .

x,

A second key result for our study of p-algebras is the following theorem bvllich is due to Hochschild (1551). Theorem 4.1.5. If K is a purely ir~separableextension of F then the map [A]-i [ A K ]of B r ( F ) into Br(K) is surjective.

Proo~f.\tie show first that it suffices to prove the result for [K : F ] < m. For, let B be a central simple algebra over K and let (x,) be a base for B / K with multiplication table xixl = C y i j k x k .Let KO be the subfield of generated by the yiZk. Then [KO: F ] < m and Bo = C K o x i is a central simple algebra, over KO such that BOK= B.Suppose we have a central simple A over F such . [ A K ]= [ B o K = ] [B]. that [AK,,]= [ B o ]Then Kow assume [K : F ] < cc. 14e use induction on the exponent of K . Assume first this is 1. Then, by 1.8.1, we have a derivation 6 of K I F such that Const S = F. If [ K : F ] = pe tllerl 6 is algebraic of degree pe over F and its minimum polynomial has the form f (A) = Ape +Y1 Ape-' . . + i e A , y, E F (Lemma 1.5.2). Now let D be a central division algebra over K . We wish to construct a central simple algebra A / F such that A K D. For this it suffices to construct a central simple AIF containing D as subalgebra and satisfying A D = K . For then AK D will follow from Theorems 4.10 and 4.11 of BA 11: By 4.10 we shall have A K = D and by Theorem 4.11. K ~ A' F M,.(DO) so AK = K 'BFA MT(D) D . To construct A we extend 6 to a derivation S of D I F . This can be done as a consequence of Theorem 1.8.3 (p. 32 ). By Lemma 1.5.3, f (6) is an inner derivation id where 6d = 0. Now form the ring D [ t ;61 of differential polynomials over D defined by the derivation 6.

-

"

+.

"

4.1. The Froberlil~sNap and Purely Inseparable Splitting Fields

+

157

+ +

The center of D [ t :61 is F[z] where z = t p e 71t~e-1 . . . yet - d. Now let A = D [ t ;S]/D[t:612. B y Theorem 1.5.4. A is central simple over F and D is a subalgebra. hloreover. the proof that the center of A is F shows that K is the centralizer AD. This proves the theorem for K of exponent 1. In the general case, we have K = K1 > K 2 > . . . > Kk > Kk+1 = F where Kt is of exponent 1 over Kt+1. The map of Br(F) into Br(K) is the cornposite of the maps B r ( F ) -+ Br(Kk).. . . , Br(K2) + Br(K1). Since K,/K,+l is purely inseparable of exponent one each of these is surjective. Hencc B r ( F ) + Br(K) is surjective. Our first application of the foregoing two theorems is

Theorem 4.1.6. B r ( F ) is p divisible for F of characteristic p.

-

Proof. In the algebraic closure F of F let F1/p = {a E F I a p E F ) . Then the map 13 --. ,3P of F ' l p + F is an isomorphism. The conlposite of this map with the injection of F into F1/? F F1/p * F gives the p-power endomorphisn~ of F . Correspondingly, by the functoriality of the map F --t B r ( F ) , we have the maps of Brauer groups

Since F l i p + F is an isomorpllism B ~ ( F ' / P ) + Br(F) is an isomorphism. Since F'/"/F is purely inseparable. E is surjective by Theorem 4.1.5. Hence the composite map B r ( F ) + Br(F) is surjective. Since this corresponds to the p-power map in F it is the Frobenius map [A] --, [A(p)].But. by Theorern 4.1.2 this is identical with [A] --, [Alp. Thus the latter is surjective and so B r ( F ) is p-divisible. For q =\ p k 2 1. we define p1/4 inductively by F1/ph = ( ~ l / p ~ - ' ) ' / p where F1/p is as before. Then the map --, 8 4 is an isomorphism of ~ ' 1 4 onto F and F1/"s purely inseparable of exponent q over F . The argument used in the proof of Theorem 4.1.5 can be used to prove the following

Theorem 4.1.8. Let A be a p-algebra. T h e n there exists a q = pk such that F~/"s (1 splitting field for A. AJoreover, zf k is minimal for this property t h e n p % ~the exponent of [A].

Proof. We have the injection F F1/4 and the isomorphism a: --, a p k of k . F1/q -+ F . The composite is the map a --, a p In F. Hence we have the maps of Brauer groups Br(F) A ~ r ( ~ ' / + q ) Br(F) (4.1.9) where

E

is [A]

-i

[AF~,~ and ] B~(F"Q) -+ B r ( F ) is an isomorphism. The

] Frobelzius rnap [A] --i [ ~ ( p ) ] . composite is the k-th power [A] --, [ ~ ( p )of~ the By Theorem 4.1.2. this coincides with [A] --i [ A I ~ 'Hence . [Alpb is obtained by applying the isomorphism B I ( F ' / ~ )+ B r ( F ) to [ A F ~ , ~It] .follows that if

158

IV. p-Algebras

g = pk is the exponent of [A] then AFIl4

N

1 and conversely if AFIl4

[A]P' = 1.

-

1 then

If E is a splitting field for A then it is clear that there exists a finitely generated subfield E'/F that is also a splitting field. Hence if E/F is an algebraic splitting field then there exists a finite dimensional subfield E1/F that is a splitting field. This and Theorem 4.1.8 imply the rnairl result of this section: Theorem 4.1.10. A n y p-algebm A has a finite dirnen,sional purely inseparable splitting field K / F and if k is the minimal exponent for such fields then pk is fhe exponent of A.

4.2. Similarity to Tensor Products of Cyclic Algebras Wie recall the Artin-Scllreier construction of cyclic fields of degree p over a field F of characteristic p: Any such extension has the form F ( v ) where vp = v a. cr E F . and a @ P ( F ) , where P(O) = OP - P. Conversely, if a E F and cu @ P ( F ) then F [ X ] / ( X P - X - a ) is cyclic of degree p over F . Hence such extensions exist over F if and only if F # P ( F ) . Note that P ( F ) is a subgroup of the additive group of F and the condition F # P(F)can also be stated as F/P(F)# 0. On the other hand, we have

+

Proposition 4.2.1. If F/P(F)= 0 then everyp-algebra over F is split (equivnlen,tly Br,(F) = 1).

Proof. Suppose there exists a non-split p-algebra A I F . By Theorem 4.1.10 there exists a finite dimensional purely inseparable splitting field K I F for A. Then we have a chain of subfields

Qp

-

+

-

Since A 1 and AK 1 we can choose where A; = I{tti,_l(Q,), 6 L = K J p I SO that AL 1 but ALl 1 for L' = K J . Then AL is a non-split p-algebra over L with a purely inseparable splitting field L' = L(0) of degree p. Then AI, -- D where D is a central division algebra of degree p over L and D contains an element u @ L such that UP E L. By the easy characteristic p part of Theorem 2.11.12 (see p. 72). AL has a cyclic splitting field of degree p ovrr L. This has the form L @ F Z where Z is cyclic over F of degree p (Theorem 8.19 of BA 11. p. 492). The existence of such an extension is ruled out by the hypothesis that F / P ( F ) = 0.

+

In the proof that a central division algebra A of degree p is cyclic if it contains an element u such that u @ F but U P E F (see p. 72), we showed that A is generated by u and a second element v such that P ( v ) = a, cu @ P ( F )

4.2. Similarity to Tensor Products of Cyclic Algebras

159

+ +

and uv = ( V 1)u.Then A is the cyclic algebra ( E , a,P) where E = F ( v ) and 01; = v 1. It will be useful to extend these results to arbitrary central simple A of degree p. Then A is either a division algebra or A = iWp(F). Hence we need to consider only the case A = M p ( F ) Suppose A contains an element u whose minimum polynomial is XP - P where we allow /3 = 0. It is a well known result of linear algebra that the matrix u is similar to the matrix el2 e23 . . . eP-l,, Bepl. Hence by applying an automorphism of Alp(F) we may assume u = XI;'-' e,,,+l Oepl. Then if we take v = C(7 z)e,, for any 7 we shall have uv = (v l ) u and TJP- v = a where a = P ( y ) . Also it is readily seen that XP - X - CY is the minimum polynomial of v and hl,(F) is generated by u and v. Thus if A is central simple of degree p over F and A contains an element u with minimum polynomial XP - a then A is generated by u and a second element v such that UP - v = cu and uv = (V 1)u. We shall now write A = [a,p) if A is central simple and is generated by two elements u and v satisfying

+ + +

+

+

+

+

+

up = p. up = v

+ a, u z ~= (v + 1)u (or [uv] =

IL)

(4.2.2)

Then the degree of A is p and the minimum polynomials of u and v are XP - P and XP - X - a respectively. In the case in which A is a division algebra E = F [ v ]is a cyclic field and A = (E,a. P) where a v = v 1. AIoreover. /3 is not a norm of an element of E . Albert's main results on the structure of p-algebras depend on the following extension theorem for cyclic fields.

+

Theorem 4.2.3 (Albert, [37], p. 194f). Let E be cyclic of degree pe, e 1 , over F of characteristic p with Gal EIF = ( a ) . T h e n E contains a n element /3 such that TEiF(P) = 1 and if P is such a n elem,en,t then there exists a n cu E E such that

>

T h e n XP - X - a is irreducible i n E[X] and if is a root of this polynomial t h e n El = E [ y ]is a cyclic field of degree p e f l over F and any such extension carL be obtazned i n this way. jf/

Proof. Since EIF is separable it contains an element /3' such that TEIF(P1)# 0. Then B = TEIF(P)-l/3' satisfies TEIF(B) = 1. Let /3 be any such element. Since TE/F ( B ) =

c:~ at(@),

Then TEIF(OP - 0 ) = 0 and so by the additive analogue of Hilbert's Satz 90 (Theorem 4.34 of BA I , p. 299) we have an a such that (4.2.4) holds. Now XP - X - a is either irreducible in E [A] or a = P ( 7 ) = 3p - y for y E E (Lemma 2, p. 675 of BA 11). In the latter case

160

IV. p-Algebras

i n E [ z ] .Applying a we obtain

O n t h e other h a n d , we have

( b y 4.2.4). T h u s D

+ 7 is a root o f X P

-

X

-

a a and hence, b y (4.2.6).

+

) 0 contrary for some i = 0 , . . . , p - 1. T h e n P = a? - 7 L and T E I F ( P= t o T E I F ( 6 )= 1. T h u s X P - - a is irreducible, and i f y is a root o f this polynomial t h e n E [ y ]is a field o f degree p over E and [ E ( - y ): F ] = pe+'. B y (4.2.8), B 7 is a root o f X P - X - a a . Hence we can extend a t o a n automorphism a' o f E ( - ! ) such t h a t

+

T h e order o f CJ'is either pe or pe+l. W e have a"? = y Hence C J = ' 7 ~ +~ T E~I F B= 7 + 1.

+ ,!3 + CJP+ . . . + C J ' - ' , ~ .

T h u s alpe # 1 so t h e order o f D' is pe+l. LIoreover, t h e orbit o f 7 under ( a ' ) contains pe+' elements. Hence t h c m i n i m u m polynomial o f 2 over F is o f degree pe+l. T h e n E' = F ( y ) is cyclic o f degree pe+l over F . Conversely, let E' b e an extension o f E t h a t is cyclic o f degree pe+l over F . Let Gal E ' I F = ( a ' ) and let a = a' E. Since E ' I E is cyclic o f degree p, E' = E ( 7 ) where t h e m i n i m u m polynomial o f y over E has t h e f o r m X P - X - a , o E E. Put T = d p e . T h e n Gal E ' I E = ( T ) . Since t h e roots ofX"-A-aarey+i, O
-

+ +

+

+

+ +

+

+

1 . 2 . Similarity to Tensor Prodl~ctsof Cyclic .4lgebras

161

We prove next the following

Theorem 4.2.11 (Teichmiiller [36]). Let K be a finite dimensional purely i~aseparahlesplitting field of ihe p-algebra A I F . Suppose K = K f ( u ) where "u K ' and K' is a su,bJSeld of K I F th,at does not split A. Let A" - CY be th,e m,inimum polynomial of u over F . T h e n there exists a cyclic a1yeb1.a ( E .a. a - l ) of degree pn over F such that A % p ( E , a, a-l) is split by K'. Proof. By replacing A by a similar algebra we may assume that K is a. subfield of A of degree the degree of A (Theorem 4.12 of BA 11. p. 224). Consider the centralizer A"'. This is central siniple over K ' and has K = K f ( u ) as maximal subfield. B y Theorem 4.11 of BA 11, A"' -- A", in B r ( K f ) (see p. 224 ). Since AK, # I , AK' 1 and since the degree of A K ' / ~ 'is p: this is a central division algebra over K'. Since A"' 3 u,A"' = [b:uP)/Kf where b E K'. We claim that we can choose the second generator v E AK SO that b = !E F . We have UP = v + b and we can replace v by vl = v b to obtain P ( v l ) = b". Since K f / F is purely inseparable, a finite number of such replacements yiclds a v = 7:k such that P ( v ) E F . Then F ( v ) is cyclic of degree p over F . By Theorem 4.2.3, the cyclic field F ( v ) can be embedded in a cyclic field E I F of degree pn. We can choose the generator a of Gal EIF so that av = v + l and form the cyclic algebra ( E , a, a ) / F with generator w over E such that

+

+

We claim that

-

( E ,a, a ) ~ '[p,uP)/K1. Since F(uP) is a subfield of K' it suffices to show t,hat

(4.2.13)

Now we have a n isomorphism of F ( u ) / F onto F ( w ) / F such t,hat u --i w. Hence It is we can replace F(uP) by F(wP) and then ( E , a, a ) F ( w P ) - ( E 3 a, readily seen that this centralizer is generated by ul and v for which we have the relations wv = (v + 1)w; P ( v ) = p . (4.2.15) Thus (E,a, [,B,wP)/F(wP). Then ( E , a: a)F(,P) and hence (4.2.14) and (4.2.13) hold. We now have

Hence K ' splits A @ F ( E , a, a - l ) .

-- (wp: P]/F(wP)

I?

From now on we assume that P ( F ) # F. Otherwise, by Proposition 4.2.1, Br,(F) = 1. The hypothesis F ( F ) # F implies that there exist cyclic extensions of degree p over F. Hence for any pe tjhereexist cyclic extensions of degree

162

IV. p-Algebras

pe over F . A consequence of this is that any A l p (F)is cyclic. This can be seen by taking the regular matrix representation of a cyclic field E / F of degree pe. This gives an imbedding of E / F in Mpe(F).Then Alp. (F) = (E.D, 7 ) . We 5hall assume also that F is infinite. This is justified since Br(F) = 1 for finite F. Theorem 4.2.11 car1 be used to prove that Br,(F) is generated by cyclic algebras (that is. by [A] for cyclic A). This is equivalent to Theorem 4.2.16. A n y p-algebra A is szrnilar to a tensor product of cyclic algebras of degrees powers of p.

Proof. We use induction on the minimum dimension of purely inseparable splitting fields for A. Let K / F be a purely inseparable splitting field of minimum dimension. The result is clear by the foregoing remark if K = F. Hence assume # F. Then K contains a subfield K 1 / F such that K = K 1 ( u ) , u K ' , up E K'. Then [K' : F ] < [ K : F ] so K' is not a splitting field for A. By 4.2.11, A -- Al @ F A2 where Al is cyclic of degree pn and A2 is split by K'. Then the induction hypothesis implies that A2 is similar to a tensor product of cyclic algebras of degrees a power of p. Then A also has this property. An important special case of this theorem is Theorem 4.2.17. If A i s of exponent p tiLen A is slrnilar t o a t e z s o r product of cyclic ulgebrus [ a ,P ) .

Proof. B y Theorem 4.1.15, the hypothesis is equivalent to: A has a purely iriseparable splitting field K of exponent 1. As above, we may assume K # F and K is of minimum degree. Then K = K 1 ( u )where u p E F . Then in the foregoing argument n = 1 and Al is cyclic of degree p. Then the argument slzows that A is similar to a tensor product of cyclic algebras of degree p.

4.3. Galois Extensions of Prime Power Degree The Galois group G of such an extension is a p-group (of order pe). For a given p-group G. Witt [36] has given necessary and sufficient conditions on F of characteristic p for the existence of a Galois E / F with Gal E / F E G, arid he has determined all such extensions and enumerated their isomorphism classes. In this section we shall prove the part of his results that we require. Let G be a p-group and let G" be the subgroup of G generated by the commutators and pth powers of the elements of G. Then G* a G, GIG* is elrmentarv abelian and G" is contained in every normal subgroup H of G sucti that G / H is elementary abelian. It is a well known result of Burnside's that if IG/G*J = p f then G can be generated by f elements and not by fewer than f elements (see e.R Scott's [li4], p. 161).

4 . 3 . Galois Extensions of Prime Power Degree

163

Let F be a field of characteri~t~ic p and let P ( F ) = {P(cy)= ap-culcu E F). Then F is a vector space over the prime field P and P ( F ) is a subspace. so F / P ( F ) is a vector space over P of dimensionality AT. 0 5 AT m. Then \Vit,t has shown that there exist Galois extension fields E/F with Gal E / F r G if aiid only if f N where GIG*1 = pf . To prove the sufficiency of Witt's conditions we assume first that G* = 1. that is, G is elementary abelian of order p.f. Then since [ F / P ( F ): P ] = N f there exist %: 1 i f , in F such that the coset,s 7, P ( F ) are linearly independent over P . Let E f be the subfield of the algebraic closure F of F of the forin F ( c l : .. . . c f ) where P ( c i ) = 7,. Then El is an Artin-Schreier extension of F so El is cyclic over F with Gal E 1 / F = ( 7 1 ) where ~ l c = l cl+l. Now suppose f > 1 and ES-1 = F(c1) @ F . . . C%F F ( C ~ - ~ Then ).

<

<

< <

>

+

is a base for E f P l It follows from this and the fact that cp = c, c E E f P 1satisfies an equation P ( c ) = 7 E F if and only if

+ 7 , that

and for c as in (4.3.2) f -1

7=

-,, +

C 11,T2 + aP

< <

-

a.

(4.3.3)

1

P(F), 1 z f , are linearly independent, it follows that c f @ F(c1, . . . , c f - 1 ) and hence [ F ( c l , . . , c f ) : F ( c l , . . . , c f - l ] = p and F ( c l , . . . . c f ) = F(c1) 6 3 .~. . 8~ F ( c f ) Then E f = F ( c l , . . . , c f ) is Galois over F with Galois group G an elementary abelian p-group of ordrr p f . If r E G then (4.3.4) re, = C , + x , ( T ) ,1 I i f Since the

<

where X,( 7 )E

P and if r' E G then

Hence X, is a homomorphism of G into the additive group ( P ,+) and so X , may be regarded as an additive character on G. Since x , ( r ) = 0 for 1 z 5 f irrlplies r = 1 the characters X I . . . . , x f form a base for the character group. Now suppose G* # 1. Then CnG* # 1 for C the center of G and hence we have the characteristic subgroup of G consisting of the elements x E G* ilC such that XP = 1. Let H be any subgroup # 1 of this group. Then H a G and H is elementary abeliaii. Since r = G / H is of order < /GI we may use induction and assume we have constructed an extension K I F that is Galois with Galois group isomorphic to Then we shall construct the required extension E I F

<

r.

164

IV. p-Algebras

that is Galois over F with Gal Gal E I K H . IT-e have the exact sequence

"

EIF

S G as an extension of K such that

where L is thc injection and u is the carionical homomorphism of G onto T. Let r = { n } and let u, E G so a = H u , = u,H. Then the multiplication in G is determined by that in H and the followiiig relations:

I I , , ~=

hu,.

(4.3.6)

RIoreover. me have the associativity conditions

These coriditions state that if we regard H as r-module with trivial r-action then (cr. r) --i h,,, is a 2-cocycle. We have the following

Lemma 4.3.8. The 2-cocycle h : ( a ,r ) --i h,.,

zs not a coboundary.

P~oof.011 the contrary suppose there exist h, E H for a E

r such that

Then if we replace TL, by v , = h;'zi, we obtain v,v, = v,,. Then V = { v , / cr E is a subgroup of G such that G = H V and H n V = 1. Since H c C it follo~vbthat V a G and G = H x V.Since H* = 1 we have G* = V* C V. Since H c G* this gives the contradiction that H c V.

r)

"

r*

\l?e note also t,hat since H c G', = G X / H arid T I T * GIG". Now let H I be a subgroup of H so H I a G. Put G = G / H I , H = H / H 1 . ij = gH for g E G. Then G* = G 2 / H 1and 7 = G/H E r . The elements u , = fi, are representatives of the cosets of G relat,ive t,o H and usirig these we obt,aill the 2-cocycle ( 3 .T ) --i h,,,. Since the conditions on G and H carry over to G and If we see that this cocycle is not a coboundary for into H either. Let IHI = pT and let X I , . . . , X , be a base for the character group of H regarded as a group of hoinomorphisms of H into the additive group of the prime field P. Applying any X , to (4.3.7) we obtain

r

Since the action of every n on P is trivial this states that ( a ,r) --t ~ ~ ( h , , , ) is a 2-cocycle of with values in the additive group (K.+) It is a well known result that if EIF is Galois and (E,+) is regarded as a module for G = Gal E / F using the natural action, then H n ( G ,E ) = 0 for n 1. (A direct proof is readily obtained using the existence of an element p E E such

r

>

4.3. Galois Extensions of Prime Power Degree

165

that T E I G ( p# ) 0. See BAII, p. 510 for a proof of the case n = 2 for Witt vectors.) Hence there exists a 1-cochain 7, : + K such that

r

Applying 'P we obtain

which states that n a -c, t K such that

--i

P v , ( a ) is a 1-cocycle of

r into K . Hence there exists (4.3.13)

7 , ( 0 )= ( a - 1 ) ~ ~ . l i e have the following

+

Lemma 4.3.14. Th'e cosets 7, P ( K ) , 1 5 i over P .

< r , are linearly independent

P7-oof. Since we may replace the base ( X I , . . . . x,) for the character group by another base it suffices to show that if x is a nonzero character of H into P and 7 : T + K satisfies .rl(a) avrl(7) - rl(0.r) = ~ ( h , , , ) and 7 E K satisfies P r / ( a ) = ( a - 1 ) then ~ 7 @ P ( K ) . Now we may replace 77 by 7'where ~ ' ( a=) ~ ( a ) ( a - 1)6, 6 E K . Then y is replaced by 7 P ( 6 ) . Hence it suffices to show that if x f 0 then 7 f 0 . Hence suppose 7 = 0. Then, by (4.3.13) ~ ( uE)P, u E Since x # 0, x is surjective on P. Hence for any a E r there is an h , E H such that ~ ( h , )= ~ ( aThen ) by (4.3.11)

+

+

+

r.

Noxv let H1 = ker X I and put G = G/H1, H = H / H 1 as above. B y - - (1.3.15). we have -h,,, = h,h,h;:. This contradicts the result proved before that ( a . T ) * hO,, is not a coboundary.

\Ve can now define the extension E = K ( c 1 . . . . . c,) where the c, are elenlents of the algebraic closure of F(> K ) satisfying P ( c , ) = x.Then E I K is Galoir, with Galois group isomorphic to H and identifying these two groups, then the clcmcilt h E H maps the generator c, of E into c, x , ( h ) . Next let n t r = G I H . The miilimurrl polynomial of c, over K is XP - X - nlz Since

+

+

the rriiriimurn polynomial of c, q , ( u ) over K is XP an automorphism v, of E / F such that

-

X

-

n ~ , Hence . we have

We claim that we have the following relations connecting the automorphism h E H and the v,. OUT = hu,r%r (4.3.19)

v,h = hv,.

(4.3.20)

166

IV. p-Algebras

Tliese are clear for their applications to the elements of K . Hence it suffices to check that they hold in their applications to the generators c, of E I K . This can be verified directly. It now follows tha,t UH7$,is a group of automorphisms of EIF iso~norphicto G. Since /GI = IHljrl and [ E : F] = [E : K ] [ K: F ] we have IG/ = [E : F].Hence E is Galois over F with Gal E I F 2 G.

We can state the result we have proved as Theorem 4.3.21. Let F be a jield of characteristic p and let G be a p-group. G* the s u b g r ~ ~ofp G generated by the commutators and p-th powers of the elemehts of G. Su,ppose IG/G*I = p f and f F / P ( F ) I . T h e n there is a n extension field E I F that i s Galois with Gal E I F 2 G.

<

In particular this result irriplies that if F / P ( F ) is infinite then Galois rxteiisions exist for any p-group. For the applications it is important to note that the -/, satisfying (4.3.13) can be replaced by y, a, where a , E F . This yields a different extension field K ( c i , .. . . ck) where P(ci) = 7;= 7, a,.

+

+

4.4. Conditions for Cyclicity If F is a field and G is a finite group then we say that G appears over F if there exists a Galois extension field E / F with Gal E I F 2 G. If A is central simple over F then we say that G appears zn AIF if A contains a subfield E I F that is Galois with Gal E / F E G and [E : F] = deg A I F . Witt's results given in the last section give a sufficient condition that a given p-group appear over F of characteristic p. In particular. Witt's results show that every p-group appears over F if F / P ( F ) is infinite. Saltman in [77] has proved the following result.

If A is cyclic of degree pe over F then every group of order pe appearing over F appears in A I F . He has also shown that Br,(F) = 1 if F / P ( F ) is finite. Hence assuming F / P ( F ) infinite as well as F imperfect the foregoing result is readily seen to be equivalent to:

If A is cyclic of degree pe over F then every group of order pe appears in

A. Saltman's proof of this result makes use of the following theorem of Albert's.

Let A be a p-algebra of degree p" over F such that F is imperfect and F # P ( F ) . T h e n A i s cyclic i f and only i f A is split by a simple purely inseparable splitting Feld of degree p f 5 pe.

4.4. Conditions for Cyclicity

167

Saltman's proof of his theorem actually gives, as we shall see; an independent proof of Albert's theorem; whose proof by Albert [39], p. 107 is fairly complicated. The basic tool for these considerations is the study of the algebras A = [a.,0) with generators u and v satisfying (4.2.2) where XP - ,O is irreducible. We have the inner derivation i, : x * [vx] in A. By (4.2.2), i-,u = u. Hence i-, stabilizes K = F[u]. Let 6 = i-, 1 K and form the differential polynomial ring K [ t ;61. Then we have the homomorphism 77 of K [ t ; S] into A such that rl I K = lK a.nd t --i v. We can apply this to prove

Lemma 4.4.1. Let k = yo

+ y1u + . . . + yP-luP-l, n/i E F .

Pmof. By (1.3.19) and (1.3.21) we have ( t kp bp-lk. Hence applying 77 we obtain

+

Then

+ k)P = tp + Vp(k) where VP(k) =

Since SZL= 21. bk = ylu + fY2u2+ . . . + (p - 1 ) 7 p - 1 ~ ~ - 1. . . @ - l k = y l ~ + 2 P - l y ~+' ~. .~. + (p - l ) p - l ~ / p _ l ~ p --l y1u . . . 7,-1 up-l = k - yo. Hence (4.4.4) V, (k) = 7; yyp . . . $-lpp-l k - "/o. Then P(2: + k) = (v + k)p - (v k) = vp Vp(k) - v - k = p ( v ) + (yp o - yo) + . . . fi/:-lPP-l as in (4.4.2). Since P ( v ) = a we also have

+

+ +

+

+ +

+

+ + +

Evidently, v + k and u generate A and [u.v + k ] = u, up = ,O. Then, using the generators u and 2: + k we see that

The 7's in this formula are determined by the choice of the generator u of K drld the elerrlent k E K . Hence we shall put

to indicate the dependence on u and k. We can now prove Proposition 4.4.8. If F / P ( F ) i s finite then Br,(F) = 1. Proof. We nlay assume F is not perfect and hence is infinite. If Br,(F) f 1 then there exists a p-algebra of exponent p. Then it follows from Theorem 4.2.17 that there exists an A = [a.B ) that is a division algebra.

-

Since F / P ( F ) < rn we can write F = Z 8 P ( F ) where Z is a finite dimensional subspace of F over the prime field P of F . Also, since any C E F satisfies CP(mod P ( F ) ) , we rnar assume that Z C FP. For 2 E F we define an L, E EndpZ as the composite of the P-linear rnap z --i yz of Z into F with the projection of F into Z determined by the deconlposition F = Z 3 P ( F ) . Then L : 7 --i L y is a P-linear map of F into EndpZ. Consider L I FP3. Since p # 0. FV is infinite and so is its dimensionality over P. On the other hand. EndpZ is finite. Hence there exists a ?; # 0 in F such that L7p,o= 0. This states that for any z E 2,yP3z E P ( F ) . We now replace u by u,'= yu+ 1 whose rninirnuin polynonlial is AP - (.;gpp 1) and replace u by a suit,able element u' such that [IL'V'] = u' and 'P(zt1) = a' E F (see p. 166). Then wit,h a change of notation we shall have A = [a.P ) and pz = %(modP ( F ) ) for all z E Z. Now let V = {(u,k) / k g K). Then V is a subspace of FIP. By (4.4.7), V > P ( F ) and V > ,OFP. Let ( E Z.Then ( = Sp where S E F and 4C (((mod P ( F ) ) . Hence C PGP(mod 'P(F)) and since PFP C V, C E V. Thus V = F . Then, by (4.4.6). A = [0, 3).Thus A contains an element whose minimum polynomial is A" A. Then A is not a division algebra. This contradicts our hypothesis.

<

+

-

-

From now on we assume F/'P(F)is infinite and F is imperfect. We prove next

Proposition 4.4.9. Let A be a p-algebra of degree pe. T h e n the following conditions on A are equi7ialent: 1. A has a simple purely inseparable splittin,g field of degree p f 5 pe. 2. A corltairls a simple pz~relyi,nseparable subfield L I F of degree pe. 3. A contains an element u with m,inimum polynomial of the form Ape - 7,; -y E F .

Proof. 1 =+ 2. Suppose K = F ( u ) is a simple purely inseparable splitting field of degree pf pe. Then u 6 K P and hence Ape-f - u is irreducible in K[A]. It follows that K can be imbedded in a purely inseparable field L = F ( w ) of degree pe over F . Then L is a splitting field for A. It follows from the theorem on finite diinensioiial splitting fields that L is isomorphic to a subfield of A (Theorem 4.12, p. 224 of BAII). 2 + 3 is clear. 3 + 1. Suppose we have an element u f A with minimuin polynomial Ape - -y over - 7 = (Apf - 6)pe-f where Apf - 6 is irreducible in F[X].Let F . Then K = F ( w ) where wpf = 6. We claim that K is a splitting field for A. For. AK contains the element z = w - u such that zpe = 0. hloreover. zpe-' # 0 since the minimum polynomial of u over K is Ape - 7 of degree pe. The existence of a nilpotent element of index pe in AK of degree pe implies AK -- 1. Hence K = F ( w ) is a simple purely inseparable splitting field for A.

<

We can now state the main theorem on cyclicity of p-algebras.

Theorem 4.4.10. Let F be a n imperfect field of characteristic p such that F / P ( F ) is infinite and let A be a central simple algebra of degree pe over F . T h e n the following conditions o n A / F are equivalent.

169

4.4.Conditions for Cyclicity I . A h a s a szmple purely inseparable splitting field o,f degree pf < p". 2. A contains a simple p,u,rel.y inseparable su,bfield of degree p". 3. A contain,^ a n e l e m e n t u wit11 m i n i m u m polynomial of t h e f o r m 2. E F. 4. E ~ l e r ygroup of order pe appears i n A I F . 5. A i s cyclic.

X P ~-

We have seen in 4.4.9 that 1, 2 and 3 are equivalent. It is clear also that 4 + 5 and 5 + 3. Hence all that remains to prove is 2 + 4. The proof of this iinplicat,ion will be based on further properties of the algebra [a, B ) that we shall now consider. where the rrlinimlim polynoii~ialover F of u is A" - /3 and Let K = F[%I] let 7 E F and ul = u 7 so K = F['u1]. Then IL? = P yp and if we define (11, k ) . k E K, as in (4.4.7), we have

+

for k = 7 0

+

+ r l u l + . . . + ry-luy-l.

Let Vl = { ( u l , k ) k E K). We require

Lemma 4.4.12. L e t t h e 7aotations he as before and suppose F' i s a subfield of F s u c h ti~czt

(2) /3 E F' (22) F = Fi(yP) (iii) If yP E F p f (f

> 1) and [ F : F'] < p f .

T h e n F = F'

+ Ifl

.l/

Proof. P u t [ F : F'] = n, yP = 6, p f = q . Let p E F . Since Vl > P ( F ) , p = pp r ... = P"f = p4 (mod Vl). Let p = ~ i pj63, ' pJ E F'. Then pq = pY634. Since 6 E F 4 , 6 = E Q , E E F, and we have a relation E" = a j ~ 3 aj : E F'. Then 6" = ~ i -Pj63 ' where b:, = a: E F14. It follows that every power of 6 is a linear combination of I , & , . . , Sn-I with coefficients in F'4. Hence

c:-'

with oj E F'4. We show next that if 0

<j

063

< n and a E F

~> j and ~ pk ,< q then ~

= ( - 1 ) ~ a p j (mod Vl).

~

(4.4.14)

First, assume k = 1. so j < p and a E F? Evidently (4.4.14) holds if j = 0, and if 1 5 j < p then o ( p 6)j = a ( @ rp)j E Vl by (4.4.11). Hence

+

+

j

o@

5

-

k=l

(i)

(mod Vl).

IV. p-Algebras

170

I11 particular. we have a B 1

-

-a6 (mod Vl) and if we assume (4.4.14) for

< k < J = 1 , then by (4.4.14), we have aP3 = - ( C

0 = (I

-

I))

=

EL=,(:))(-i)jPk.

-

we have o~3J

0 < j < p - 1: a E F P . Now let pk > j j pp,a E where 0 5 j" < p and 0 < j' < j . Then

FP'

()

( - 1 ) ~ ~ ' a 6 ~Since ).

(-l)iob'(mod

1;)for

and write j = pj'

+ j"

(since Vl > P ( F ) ) . Since all" FP"-', ?f' E F p f - l c Fpk-'a1/py"' E Fpil and 3' <, pk-l. using induction we may assume that .r1/p7j /iJ E (-1)3'01/pyf'@ . Then I,

I

~ J'- < ~ pi-1. using induction we may Since a l / p E ~ p * - l , r Yt ~ p and assume that a1/py~"63 E (-1)3'a1/~3~"p3'(modVl). Then

and since 0 ~ 3 E' F~P , this is = ( - 1 ) ~ ' p o ~ ~ ' ~ ( - l ) Pby 3 ) )the result we established before. Then we have (4.4.14) for all 0 J < n and a E F4. We apply this t o the terms a363 in the expression (4.4.13). This gives p4 = C(-l)JaJ,OJ (mod Vl) and since p E F' and p = pq (mod Vl), p E F1+V1.

<

We can apply Lemma 4.4.12 t o prove

Lemma 4.4.16. Let A = [ a , @ )and let F' be a subfield of F such that [F : F'] = n < cc an,d F' 3 p . Let a1 be any element of F that is separable over F'. T h e n there exists a y E F and a n a' E F' such that A = [al+ a', /3 + yp). Proof Let S be the subfield of separable elements of F I F ' . Since a = a p P ( a ) . [ a . p) = [ a p 3D ) . Similarly l a p , P) = [ a p 2 ,P) = . . .. It follows that we may assume a E S. Then [a.P ) is defined over S so replacing F by S we may assume F is separable over F'. Then F = F1(p) = F1(pP)- . . . - F'(flf) f-1 . where p f > n. If we now take 7 = pp in 4.4.12 we obtain F = F' + Vl where Vl = {(u / k) / k E K = F ( u + 3)). We have A = ( a , P] = ( a * ,P + a P ] for some a* E F . By 4.4.12. a* - a1 = a' ( u l . k) and hence. by (4.4.6). [ax. .O yP) = [a1 a ' . P ?p). Then [ a ,P) = [a + a 1 /? , yp).

+

+

+

+

+

We can now give the

Proof that 2 3 4..Let K = F ( w ) be a simple purely inseparable subfield of A. Jtre have u p e = O E F . Put u = u:pP-'. L = F ( v ) . Then u p = P, [L : F] = p

4.5. Similarity to Cyclic Algebras

171

and A' = AL is central simple of degree pep' over L. We have the P-linear map a -. crQ f ( F ) of L into F / f ( F ) . It is readily seen that this induces an isomorphism of L / P ( L ) onto F / P ( F ) . Hence L / f ( L ) is infinite and so every p-group appears over L. Hence, by induction on the degree, if r is a group of order t,lien there exists a subfield Il.I/L of A 1 / L that is Galois over L with Gal 1ZdlL 2 r.Now M = S 8~ L where S is Galois over F with r.Let A" = A S . This algebra is central simple of degree p over S Gal S / F and it contains M = S ( a ) ,V P = /3. Then A" = ( a ,P ] / S where a E S , P E F . We now distinguish two cases: I . G i s elementary abelian. We take l- elementary abelian. Then S = F ( c 1 ; .. . ' c e - I ) where f ( c , ) = 7 ,E F and the cosets 7, f ( F ) are linearly independent over P. If we apply Lemma 4.4.16 with a1 = 0 to A" = ( a ,P] we see that A" = (a1:-1 where a' E F . Then we have a a in A" such that f ( u ) = a' and [ S ( v ): S ] = p. If a' P ( F ) is not a linear combination of the 7, f ( F ) with coefficients in P then S(71) = F ( c l , . . . , c,-1, v ) is Galois over F with Galois group G tha,t is elementary abelian of degree pe over F . On the other hand: if a' f ( F ) is a linear combination of the yi P ( F ) then A" is split over F . In this case we can choose -ye so that yo P ( F ) ,. . . , y e f ( F ) are linearly independent over P . Then if c, is an element such that f ( c , ) = Y ~S ,( c e ) = F ( c l , . . . , c,) is a cyclic field extension of S of degree p and is elementary abeliarl of degree pe over F . Since A" = A.(T,(S) S(c,) can be imbedded in h f p ( S )so we may assume S ( c e ) c A". Then F ( c l , .. . , c,) c A" C A so the elementary abelian G appears in A / F . 11. G i s n o t elementary abelian. We can pick a cyclic subgroup H of order p contained in C n G* (notations as in Section 4.3) and we may assume that Gal S / F r = G I H . We choose a1 E S so t,hat if E = S ( v ) with f ( v ) = a1 then E / F is Galois with Gal E / F G. By Lemma 4.4.16, there exists an a' E F such that A" = (a' a l , P y p ] / S . Then A'' contains an element vl such that P(v1) = a' a l . Since a' E F it follows from the remark following tjhe statement of Theorem 4.3.21 that S ( v l ) is Galois over F with Gal S ( u l ) / F S G.

"

+

+

+

+

+

"

+

+

+

+

"

+

4.5. Similarity to Cyclic Algebras The next result we shall prove is also Albert's.

Theorem 4.5.1. T h e t e n s o r product of t w o cyclic algebras of de,qrees powers of p i s cyclic. For the proof we require the following lemma on fields.

Lemma 4.5.2. Let L be a separable extension field of a simple purely inseparable field K / F . T h e n there exists a v € L such that l q I K ( v )generates KlF.

172

IV. p-Algebras

E where E I F is separable. Then E = F(711).Let Proof. I i e have L = K f ( A ) = Am alAm-I . . . an,be the minimum polynomial of u: over F. IVe can write

+

+

+

13-1

and since f ( A ) is a separable polynomial one of the f , ( A ) # 0 for z > 0. Suppose f,(A) # 0 for r > 0 and that K = F ( u ) where the rnininlum polynomial of ZL ovpr F is A p e Now f , ((uA)P)# 0 in K[A]and since F is infinite, there exists an a E F* such that f ( ( a u ) p )# 0. Now put

-,.

The minimum polynornial of v over K is

f (A

+ a u ) = Am + . . . + f ( a u )

(4.5.5)

so XLIK(.) = ( - l ) m f (cuu,) and

This is an expression for f ( a u ) as a linear combination of the base (1.u , . . . . 2 ~ P - l ) of K = F ( P Lover ) F(up) with coefficients in F(up).Since f , ( ( a u ) " ) a r# 0.f ( a u ) @ F(uP) and hence 1 3(v) ,~ @ F(uP). This implies that NLiK ( v ) generates K I F .

Iic can now give the Proof of Theorem 4.5.1. Let Ai = (E,! D,; yi), i = 1.2, where the degree of A, = [E, : F ] = pC'. We use induct,iorl on the degree of Al. (The result is trivial if el = 0 . ) Suppose first that yl = b y , 61 E F . Then: by the inflation theorem for cyclic algebras (Corolla,ry 2.13.20). Al = (El,a1,6?) ( E ; .0;,61) where Ei is the subfield of degree of El and a; = a , El. Then -4 -- (E:. a:. 61)E F ( E 2 0, 2 ; :f2) which is cyclic by the induction hypothesis. # 6: for S1 E F. Then A p e - -,I is irreducible in F[A] Now suppose and Al contains a purely inseparable subfield K 1 of degree p e l . Now A > K 1 8~A2 = A21(1 N (K1 @ p E2,a;;7 2 ) where gh is thc cxtension of a2 to Eh = K 1 R F E 2S U C ~that a; / K 1 = I K , (Corollary 2.13.21). By Lemma 4.5.2, Eh contains an elernerlt .c' such that &;(v') generates K 1 / F . We have

-

and hence this algebra over K 1 contains an element v; whose minimum polySince A ' E ; , ~ , generates K~ , so nomial over ~1 is ~p~~ - n/21VE;lK1(~1'). does ? 2 % ; / ~(~v ' ) . Then ( ~ ~ N E ;(/vK' ), ) ~E~ F' and this is not a p-th power in F . \Ve have u!-jpL2= n 1 2 ~ E ; i K 2 ( ~ and 1 ) (11')

4.5.Similarity to Cyclic Algebras

173

Slnce this is not a p-th power in F . X p e l f e 2 - (12~TE;lK2(~'))Pe1 is irreducible in F[X].It follows that F[ub] is a sirnple purely inseparable field of degree pelfe2 contained in A1 @?F A2. Then. by the criterion for cyclicity (Theorem 4.4.lo), Al Z F A2 is cyclic.

Tile foregoing theorem and Theorem 4.2.17 imply the main theorem on the structure of p-algebras:

Theorem 4.5.7. A n y p-algebra zs szmzlar t o a cyclzc algebra.

'1%-eclose this section by proving an addendum t o Theorem 4.4.10: Theorem 4.5.8. T h e ,following conditions o n a c e n t ~ a lsimple algebra A of degree pe are equivalent: (2) A h a s a purely inseparable splitting field K of degree pf pe, (iz) A i s cyclic, (iii) A contains a szmple purely inseparable su,bfield of degree pe (cf. J. M y m n , Hood [71]).

<

Proof, I n vicw of 4.4.10 the only implicat,ion t,hat has to be proved is (i)+ (ii). \h7e USP induction on e. Tlle result is clear if e = 1 b y Theorern 4.4.10 1. and it is clear also if e > 1 and f = 0. Hence we assume e > 1 and f \17e clairn t,hat A contains the field K: \lie have A = iI'fPs( D ) where D is a central division algebra of degree ph and e = g + h . By the basic theorern on finite dimensional splitting fields (Theorern 4.12, p. 224 of BA 11): pf = p"h so ,f = X: + h and K is a subfield of AifP',i, (D). Since f e , k g and hence K is a. subfield of ?VIPs( D ) = A. Now K contains a subfield F ( u ) where u @ F arld UP E F.Then [ F ( u ) : F] = p. Let B = AF(7L). Then B is central simple of degree p f p l over F (11) and K I F ( u ) is a splitt,ing field for B / F ( u ) of degree p f p l over F ( u ) . Hence by the induction. B = (E.D, -y)/F(u) where E is cyclic of degree over F ( u ) . Let u:E B be a generator of B over E such that w a = ( a a ) w and wpe-l = 7 E F ( u ) . Since [(E,a! 7): F ( u ) ] = p2(e-1), the lninil~lurrlpolynomial of TL! over F ( u ) is P e - l - y.We distinguish t,wo cases: I. 7, $! F. Thcn the minimum polynomial of 11) over F has the for111 A"' -0. Then A is cyclic by Theorem 4.4.10. 3. (p. 167) 11. y E F . I r e have E = E' XF F ( z L )where El is cyclic of degree pep' over F with Galois group ( 0 ' ) u~herea' = a E' (Theorem 8.19. p. 492 of BA 11). The subalgebra C of A generated by E' and 'u; is cyclic of degree over F. \;Ye have A = C 8~AC and AC is central simple of degree p. Since AC contains F(IL)it follows that AC is cyclic. Thus C and AC are cyclic and helice A is cyclic by Theorem 4.5.1.

>

<

<

IV. p-Algebras

174

4.6. Generic Abelian Crossed Products In the next section we shall give a n example due to Anlitsur and Saltman of a non-cyclic p-algebra. The construction of this example is based on the concept of a generic abelian crossed product that we shall now consider (Amitsur and Saltlnan [78]). Let F be a n arbitrary field: E a n abelian extension of F with Gal E I F = G. Then G has a base, that is. G = (al)x . . . x (a,). We shall fix this base once and for all and we put G , = ( a , ) ,G,I = o ( a , ) = nz Also we write G Z l...,, = ( a , l , . . . .a,,) so / G,,...,, I = n,, . . . n,,. Put

FZ,...,i, = Inv Gi ,... i,

(4.6.1)

Nil...ik = N E I F , , ,, .

(4.6.2)

and Then GI ..., = G , Fl..., = F: N1..., = k l F Any . a, stabilizes Fi,...,, and if 2 { i . i } then a 1 F . ., = 1 Otherwise, the restriction of a, to F,, ...i, has order ni. For any a E E and i # j we have %,.

Also ATt1...,, ( u Z a )= a , Nil .. . i k ( a )

(4.6.4)

and if i E {il, . . . , i k ) t>hen

(4.6.5)

N,, ...i, (a,a) = N i l . ..i, ( a ) .

Let A = (E.G, k ) . The automorphism a , can be extended to a n inner automorphism I,$ of A where z, is determined up to a multiplier in E*. Then we have z,a = (a,a)z,, a E E (4.6.6) and since the automorphism I,% I,, I&'I;' = I z % , 3 , _ ,,-I we have z,z,z,lzY1

= utg E

E * . or,

is the identity on E l 3

Similarly, since 0:' = 1,

zp' = b, E E * . We have the following relations connecting the ut3 and the b,:

(4.6.8)

3.6. Generic Abeliarl Crossed Products

175

Pioof. (4.6.9) is clear. For (4.6.10) we apply IzJ to (4.6.8) to obtain

since we have the formulas

for a E E,bv induction on k . By (4.6.10) we have N j ( o j b i )= N j i ( u j i ) N 3 ( b , ) . Hence fV,(bi) = N l i ( u j i ) N j ( b i ) and since N j ( b , ) # 0 we have (4.6.11). To prove (4.6.12) we apply IZ3to zizk = uikzkzi. This gives

I,, ( z i z k ) = ( z j z i ~ l ) ( z j z k z ; l= ) (w.,izi) ( u j k z k ) =~

Iz,

, , ( ~ " J k ) ~ z ~ k

( u z k ~ k ~= i )( g j ~ i k ) ~ j k ( ~ k u j i ) ~ k ~ i

= (ojuik)ujk(okuji)ukizizk

Hence U ~ , ( ~ ~ = P L( a~l ~~ ~) , k ) u , k ( a k ~ ~ , , )B u yk Z(4.6.9), . this gives (4.6.12). We write A = (E,0. U , b) where cr = ( a l . . . , a,). U = (ZL,~),b = ( b l , . . . b,). We have noted that the z , are determined up to multipliers in E*. Suppose z, is replaced b y z; = c,z,, c, E E*. Then it is readily seen that and b, are replaced by uij and bi where

Thus ( E . a , U.b) = ( E ,o, U'. b') where U' and b' are given by (4.6.14) and (4.6.15). VCTeshall show that given any r x r matrix U = (u,,) with u,:, E E* and any b = (b,). b, E' E* , such that (4.6.9)-(4.6.12) hold then there exists an abelian crossed product ( E ,G, k ) = ( E ,0 , U , b). First we drop the b, and we construct an iterated twisted polynomial extension E [ t ;a. U ]where t = ( t l ,. . . , t,), a = ( 0 1 . . . . . a,). U = (uz,)as follows. Let a(" = (((T.. . . , o k ) , t ( k ) = ( t l , . . . ,tic).u(')= (u,, / 1 < 2 , 5~ ~ c )so D ( ~= ) a . t ( ' ) = t. u ( ~ =)U. u(')= ( 1 ) . We define ~ [ t ( a l () ' ;) . u(')]as the twisted polynomial ring E [ t l :011 whose elements can be written in the form

where

+

+ +

tla = (ala)tl, a E E

(4.6.17)

and a0 a i t l . . . a,ty = 0 H every a, = 0. We recall that E [ t l ;all is a domain and it is readily checked that if 77 is a homomorphism of E into a ring A and there exists a zl E A such that

176

IV. p-Algebras

then

is a hoinoinorphisrn of E [ t l :a11 into A extending r/ and mapping tl into z l . More generally. wc have Proposition 4.6.20. Let R[t;a ] be a twisted polyn,omial ring defined b y a ring R and an automorphism a of R. Then:

(i) R[t:01 is a domain if if is a domain. (ii) Ifrj is a h,omomorphism of R into a ring A and A contains an element z suclz that z ( q a ) = ( q a a ) z , a 6 R, then there exists a unique extension of 77 to a homomorph,ism of R[t;a ] into A su.ch that t --i z . The proofs are clear (cf. Section 1.1). Now suppose wc have constructed a ring ~ [ t (a("), ~ ) u(')] ; with the fol!owing properties: (i) ~ [ t ( "a('"); ; u("]contains E as subring and is generatled by E and t(') = ( t l ;. . . , t k ) such that

aEE, l
ail ...i, E E and the t's are algebraically independent over E in the sense t,hat C a,,... i,t';l . . . t: = 0 =+ every a ,,...,, = 0. (iii) If A is a ring containing E as a subring and 17 is an automorphism of E and zi: 1 i k , are elements of A such t,hat

< <

the11 q has a unique extension to a homomorphism of E [ t ( " ; a("), u("] into A such that t, --t z,, 1 5 i k . (iv) E [ t ( l C )a;( ' ) , u('")]is a domain.

<

+ u('+')]having these Assuming k < r we construct a ring ~ [ t ('1;~a("'). properties wit,h k replaced by k 1. Let 77 be the autonlorphism ak+l of E and let zi = 7Lk+l,,ti. Then (4.6.23) holds since for a E E ,

+

3.6. Generic -4belian Crossed Products

177

by (4.6.9) and (4.6.12). Hence we have a ring elidomorphisrrl Sk+1 o f E [ d k ) a(" ; . u ( ~ )such ] that

. . . t: where b,,... i , is T h i s m a p s C a,,..., ,t? . . . t: int>oC a k + l ( a i,...,, )b,,..., a product o f uk+l,, and their conjugates. Hence ak+1 is surjective and b y t h e algebraic independence o f t h e t's it is injective. Hence irk+l is a n automorphism ,; ( ~so) we ] can form t h e twisted polynomial ring o f ~ [ t (d~k )u Since ~ [ t (a ~ ( k ) :u: ( ~contains ) ] E and. is generated by E and t ( k ) , ~ [ t ( ~ + ' ) u ( ~ + is ' ) generated ] b y E and d k f l ) . We have tk+1a = (i?k-la)trc+~?tkilt, = (i?k+lti)tk+l = uk+l.,tith Hence (4.6.21) holds for 1 i, j k 1 so condition ( i ) holds. A straightforward verification shows that (ii) holds. Now let r j b e any automorphism o f E and A a ring containing E as subring and containing elements z,, 1 5 i 5 k + 1 , such tha,t (4.6.23) holds for a E E and 1 5 i , j 5 k + l . T h e n r j has a unique extension t o a homomorphism 7 o f E [ d k ) ; u("] such t h a t t, * z i , 1 5 i 5 k . By Proposition 4.6.20 (ii)' this has a unique ~ x t e n s i o nt,o a homomorphism o f ~ [ t ( " l ) ; U(lc+')]such that t k + 1 * zk+b Hence (iii) holds. Finally, ( i v ) follows from Proposition 4.6.20 ( i ) . T h i s completes t h e inductive step o f t h e proof o f

<

< +

Theorem 4.6.26. Let E be Galois over F ,u~ithabelian Galois group G = ( a l ) x . . . x (a,.) and let U = (u,,) where uij E E* and (4.6.9), (4.6.11); (4.6.1.2) hhold. Th,en we can construct a dorr~ainE [ t ;a , U ] = ~ [ t (a(,.), ~ )u; ( ~ ) ] containing E and generated by E and t l . . . . 't,. such that (4.6.21) holds for a E E and 1 5 i ! j < r . Moreover, e71el.y element of E [ t ;D. U ] can be written i n th,e form Z a ,,... i F t y. . . t p where ui ,... iT E E and the t ' s are algebraically independent over E . Finally, E [ t ;0:U]h,as the universality property that if r j zs a n a;u,tomorph,ism of E and A is an extension ring of E containing elenlerlts z, such th,at (4.6.23) lzolds then r j has a unique extension to a homomorph,ism, of E [ t ;a , U ] into A such that t , * s i , 1 < i < r . I n addition t o t h e matrix U we now suppose we have t h e vector b = ( b l , .. . ' 0,)' b, E E* such t h a t (4.6.10) holds. Lct I(b,) b e t h e left ideal i n E [ t ;(7, U ] generated by ty" bbi(n,= ( a , ) / ) .T h i s is also a right ideal since (tn7 - b,)a = (a:'a)ty3 - ab, = a(tn'

-

b,)

178

IV. p-Algebras

for a E E and

since ayll?li, ( u i J )= bi' ( a ~ l b , by ) (4.6.9) and (4.6.10). Hence

+

T I(b,) and write I = x I ( b ) for x E E [ t :a . U ] .Since a --i a, a E Let I ( b ) = C E . is a monomorphism we rnay identify a with Si and regard E as imbedded = b, every elerrlent of E [ t ;a. U ] can in E [ t ;a , U ] = E [ t ;a , U ] / I ( b ) .Since be expressed in the form

cz

Hence regarding E [ t ;a , U ] as left module over E we have

[ E[t;a , U ] : E ] < n, [E[t;a , U ] : F ] 5 n2

cT

(4.6.28)

<

and equality holds if and only if the monomials . ..:f 0 z l 5 n3. are linearly independent over E . Now suppose that U and b are determined as above from a set of z, of an abelian crossed product A = ( E ,G. k ) = ( E ,z , U , b). Then we have the homomorphism q of E [ t ;a . U ] onto A such that n * a.t, --i z,. Since I ( b ) c k e r q and [ A : F ]= n 2 = [ E [ t ; a , U ] / k e r qF: ] [ E [ t ; a , U:] F ] it follows that I ( b ) = ker 77 and the induced map of E [ t :a . U ] into A is an isomorphism. There are several important consequences of this result.

<

Theorem 4.6.29. (i) ( E ,a: U , b) that (4.6.14) and (4.6.15) hold.

" ( E ,a , U ' , b')

H

sn2,

there exist ci E E* such

(zi) If z, and zi, 1 < i 5 r ; are generators o,f A = ( E ,a , U, b) over E that determine the same matrix U and ziector b then there exists an automorphism 77 of A , that is the identity on E a,nd maps zi * z,', 1 < i < r .

"

Proof. (i) Suppose A = ( E ,a , U. b) A' = ( E ,a , U ' . b'). Then there exists an isomorphism 7 of A onto A' that is the identity on E . Then we have ( q z , ) a = ( a , a ) ( v z , ) . a E E , and (vz,)(qz,) = I L , ~( q z J ) ( v z , ) .( q ~ , ) ~ -b,. The first of these implies that (772,)-1z: centralizes E . Hence there exists c, E E* such that zi = c,(qz,). Then the other two conditions imply (4.6.14) and (4.6.15). Conversely. suppose we have c, E E* such that (4.6.14) and (4.6.15) hold. Then. by replacing z, by c,z,, 1 5 z 5 r , we may suppose U' = U and b' = b. We have seen that there is an isomorphism of E [ t :a. U] onto A such that a * a. a E E , & 6 z,. Similarly. we have an isomorphism of E [ t ;a , U ]

4.6. Generic Abelian Crossed Products

179

onto A' sucll that a --t a . 6 --i 2,'. Then we have an isomorphism of A onto A' such that a --i a and z, zi. (ii) This is clear on specializing the foregoing result to the case A = A'. \Ire can use Theorem 4.6.29 (ii) to prove the following generalization of Hilbert's Satz 90. Proposition 4.6.30. Let E be a n abelian extension of F with Gal E I F = ( a l ) x . . . x (0,) and let c = ( e l ,. . . , c,), c, E E * . Then there exists a n a E E* such that C, = a ( a i a ) - l (4.6.31)
N i ( c i ) = l , ( a i c , ) c j 1 =(o;.c,)ccl.

(4.6.32)

Proof. Direct verification shows that if c, = a(a,a)-I then (4.3.32) holds. To prow the converse we consider the algebra A = EndFE. It is known that A = E G = CgEG E o where E is identified with the set of left multiplications ~ ( ( c a ) L u ) for a t in E (BAII, p. 473). Since a a = ( o a ) a (that is. a a = E, A = ( E ,G , 1 ) and we may take z, = a , to obtain A = ( E , a ,1. I ) . Now . 1. Hence, put zi = c,z,. Tllen (4.6.32) implies that zizl, = z;zi and ( z : ) ~= by Theorem 4.6.29 (ii) there exists an automorphisrn of A that is the identity on E and maps z, into z:, 1 1 5 r . This has the form Ia. a E E*. Hence z: = e L z L= a z 2 a p 1= a(a,a)-'2% and c, = a ( a , a ) - l .

<

We consider again E [ t ;a . U ] and we determine the center of this domain. Theorem 4.6.33. There exist a, E E * such that the center C of E [ t ;a , U ] has the form F[$1;. . . :<,I where $, = aitYt. Moreover,, the <,are algebraically in,depen,derst over F .

Proof. The elements t y ~ e n t r a l i z eE since 0%'= l E . Also we have seen that tryt, = hT,(u,,)t,t:%. Now put c,, = N , (u,,). Then if we apply N k to both sides of (4.6.12) a simple calculation using (4.6.9) and ATk( a k a )= Nl,( a ) shows that ( o a c 7 k ) c ; j = ( ~ , c , ~ ) c ; ' .Also AT,(c,k) = N,Nk(utic) = Nzk(u,l,) = 1. by ( 3 6.1 1) Hence. by Proposition 4.6.30, there exists an ak E E* such that c,k = a k ( a , a k ) - l . 1 z r . Then ( a k t : k ) t , = t , ( a k t I k ) so El, = a k t r k E C . The algebraic independence of the t, over E implies the same result for the $, over E and hence over F. It is clear also that every element of E [ t ;a, U ] can be written in one and only one way in the for~n

< <

xA,,...,rti'. . . t > ,

o < i , < n,,

A?,... i , E E [ <, .~. . ,<,I.

Since

t 1Z 1 ..tiyn . T = ( a1i l . . .o>a)t";' and the autornorphisms ail it follows that

..t>

. . .aFr are distinct for the i, in the indicated range.

180

IV. p-Algebras

a ( xA

't . . . t j r ) =

(1 A .".t;' ,,..

. . . t:)a

implies that A,, , r z = 0 for all (i) except ( 0 , . . . - 0 ) . This implies that the Since at, = t , n for 1 5 I r centralizer of E in E [ t :a , U ] is EIE1.. . . . implies a E F it follows that C = F [ c 1 , .. .

.I,[

<

,<,I.

SVe return to the problem posed earlier: Given the abelian e~t~ension EIF with Galois group G = ( a l ) x . . . x (cT,), matrix U and vector b such that (4.6.9)-(4.6.12)hold, t,o co~lstructan abelian crossed product (E,a, U, b). We shall show that E [ t ;a ! U ] = E [ t ;a , U ] / I ( b )is the required algebra. Since I ( b ) is the ideal generated by the elements tr' -hi and t" = a;'[,, I ( b ) is generated by the elements a;'[i - b,. Hence we have = a,bi ( 2 = z I ( b ) ) . It follows that every element of E [ t ;a . U ] is an E-linear combination of the n-elements . . . t?, 0 5 ij < n,, . -.We claim that these are linearly independent over E. . . . t> = 0 , a Suppose C a , , . . . , T E E . Then we have C a , , ...,v t;' . . . t p = ~ . f (a,'[] j 11,) for f,, E E [t;a . U ] . Then fj is an E-linear combination of the base ( t y . . . t : [ : ' . . . [ ? , 0 5 i j < n j $ 0 5 k j < oo). If some f j # 0 let ti' . . . tz
c,

+

ti'

-

.T: [

(T

<

a,, ...,r = 0 and hence that . . .;? 0 i i < n,,)is a base for E [ t ;C T , U ] over E. -. SVeliave t 2 1 . . . e a= ( a ~ l ~ ~ . r r > a ) $ . SinceG . . ~ . = ( c r i l . . . a $ 10 5 ij < n J )\ire see that E [ t ;a , U ] is a crossed product (E,G , k ) for some factor set k . Hence this is an abelian crossed product (E,a . U,b). We have therefor proved Theorem 4.6.34If EIF is abelian with Gal EIF = G = ( 0 1 ) X . . . x (cT,.): a = ( a l , .. . ,a,.) and U and b is a matrix and vector of elements i r ~E * satisfyir~g ( 4 . 6 9)-(4.6.12) then there exists a n abelian crossed product ( E ,a ! U,b ) . We have seen that for ally U satisfying (4.6.9). (4.6.11)-(4.6.12) we can construct E [ t ;a . U ] as an iterated twisted polynonlial ring which is a dornain with center C = F [ [ l .. . . . [,I, [, = a,trq algebraically independent over F . TVe now have

Theorem 4.6.35 The central localization E [ t ; a ,UIc* is a central division algebra over F(E1,. . . , <,.) (cf. Theorem 1 .A. G).

Proof. The center of E [ t ;a . UIc* is Cc* = F([1, . . . -6,). Since

(tZ,'" ' t ~ < ~ ' " ' < , " k10 r 5 ik < n k . k j

> 0)

is a basc for E[t;CT.U ] over E if ( u k ) is a base for E / F then

4.6. Generic Abelian Crossed Products

181

.

is a base for E [ t ; a: U] over F. Hence E [ t ;a, U ] is a free F[[l,. . . [,.]-module with base (uktZ,l. . . t > / 1 < k n,0 ij < n:,) Then E [ t ; a ,UIc* has this base over F(c1, . . . , 6,). Then E [ t ; a: UIc- is finite dimensional over a field. Since it is a dornain it is a division algebra. It is readily seen that E [ t ; m, U]C- = ( E : a , U , a - l [ ) l F ( [ ~ : . . . ,&).

<

<

1% shall call E [ t ; a, U ] c - = (E: a, U , up'[) the generic abelian crossed product determined by the matrix U . VbTenow consider criteria on U for tensor fact,orization of generic a.belian crossed products and more generally of any ahelian crossed product ( E , 0 , U,b). We note first that corresponding t o t,he fact,orization of G as (01) x . . . x (a,) we have a factorization of E as tensor product of cyclic fields. Let E(') = Fl,,,;,,,,. Then [E(') : F] = ni and I ~ ( ' 1 hIorcover, . E = E(') Br;. . . . ~ ( ~ 1 . Gal E(')/F = (a:) where cri = GK, Now suppose U = 1. Then the z, commute and if A, is the subalgebra gener) z, then Ai = (E('), a;, bi). It follows that A = Al Z F . . .aFAr. ated by E ( Z and More genera,lly, if we take into accouiit (4.6.14) w-e see that if there exist ci E E*,1 5 i 5 r , such that

the11 we call replace U by U' = 1 to obtain (E,a. U. b) = (E.a . U', b') = Al a~. . . R F A,. where A, is cyclic of degree n,. Of particular interest is the case in which r = 2. Then if u = 2 ~ 1 2 .u2l = t1-l a i d u11 = 1 = 7122. The conditiolis (4.6.36) reduce to the single condition

15%lion7 iiitroduce some abbreviations: 1Ve write m = ( m l , . . . , m,) where the In, are non-negative integers and we write zm = znL1. . . z r r , GK" = cry1 . . . aFr.For fl = (12~:. . . . n,) we write

If H denotes the nlultiplicative group in A = ( E . a . U, b) generated by zl. . . . z, then the derived group H' is generated by the elements u,:,= Z-l -1 and their conjugates r u , , , ~E Gal E/F. Since ufiI,, E H' this ' 3 % element is a product of the uLJ and their conjugates.

.

Definition 4.6.39 The matrix U is called d~genemteif there exist m and n such that (a", a") is not cyclic and el. c2 E Ex slirh that

Otherwise. U is non-degenerate. Lemma 4.6.41 If A = ( E , a, U:b) and U i s degenerate then the exponent of A in B r ( F ) is less th,an its degree (= [E : F ] ) .

182

IV. p-Algcbras

Proof. Let E' = Inv(am,a") and A' = AE'. Then AE' is central sirrlple over E', [A : F] = [A : E1][E': F] and AE' -- AE, in Br(E1) (Theorerns 4.10, 4.11, p. 222: 224 of BA 11). If d E E' then afid = d = o n d and (c1z")d = cldz" = d(clzm) Hence x = clzm E A'. Similarly, y = CZ'Z' E A'. Also F C El. It follows from (4.6.38) and (4.6.40) that xy = yx. T h e relations [A : F] = [A' : F][E1: F] and [A : F] = [E : FI2 imply that [A' : E'] = [E : Ell2. It follows that A1/E' is an abelian crossed product of EIE' with generators o", a" of H = Gal E/E1and generators x and y over E. Since H is not cyclic, by the Fundamental Theorern on finitely generated abeliarl groups H = (0"') @ ( ~ ~where 2 ) the order dl = o ( a m l ) d2 = o(afi2) and dl # 1. Let E, = Inv om3,j # i . Then E/E1= E1/E1@E2/E' and E,/E1 is Galois with Galois group that can be identified with ( 0 " ~ ) .Hence [E, : E'] = d, and [E : E'] = dldz which is degree of A'/E1. we have 0"' = ( ~ ~ ) ~ ( o " ) ~ , om2 = ( o m ) u ( o n ) w and if we put x l = xSyt,x2 = x"yW then we have the d cyclic algebra Ai = (Ei7a"] ; :/J ) where yj = xjJ. Moreover, A1/E' = Al @ A2 and since the exponent is a divisor of the degree, Ard2 -- A? @ A? 1. Thus (AEOd2 -- 1 in B r ( E 1 ) and since A"' AE, we see that (Ad2)*/ (AEod2 1. Hence E' is a splitting field of Adz. Now it follows from the basic theorem or1 splitting fields ( p 224 of BA 11) and the fact that the exponent is a divisor of thc degree that (Ad2)IE'F ] -- I. Since d2[Er: F] < dld2[E1: F] and [E : E'] = [Gal E/E1]= dld2. d2[E1: F] < [E : E1][E': F] = [E : F]. This implies that the exponent of A in B r ( F ) is less than the degree.

--

-

-

We consider again a generic abelian crossed product ( E , a, U. a-1[) = E [ t ; a, U]c* (in the notation of Section 4.6). Suppose first that r = I. Then E[t: o, U] is the twisted polynomial ring E [ t l ; all where E is a cyclic extension of F with Galois group ( a l ) and [E : F] = n l . The center C of E [ t l : all is FIE1].El = tT1. Then E [ t l ; olIc- is the cyclic algebra (E(E1), ol,E1) where ol is the extension of al to E(E1) such that alEl = $1. It is readily seen that no (,k. 1 k 5 n1, is a norm in E ( & ) . Hence. by kvedderburn's norrn theorem. (E(E1),al, El) has the full expor~entn l (cf. p. 30). 2. From now on we assume char F = p. Suppose also that n l = pf. f For example. we can take E to be an extension of a finite field F such that [E : F] = n l and take a1 to be the Frobenius automorphism a --t aplFi.kVe can apply Theorem 4.4.10 to conclude that every group of order n l appears In (E([1), a1. El)/F(
<

>

>

+

crossed product ( E( E l ) , a , U,b ) where U =

(

y)

is non-degenerate since

thc algebra has full exponent (Lemma 4.6.1). All we require of this is

Lemma 4.7.1. For a suitable F there exists an abelian extension E / F with Galois group a direct product of two cyclic groups of order pf. > I and a u E E* such that U =

(

';) is nun-degenerate in the sense of Definition

4.6.39.

>

Now let r 2. We recall that any element of E [ t ;a?U ] has the form t y . . . tZ,., ai ,... i F E E and the monomials t? . . . t: are linearly a = Za irldrperident over E . We order the monomials lexicographically and define t,lie leading term X(a) of a as ai t;l . . . t> with a # 0 and t? . . . tZ,. maximal in the lexicographic ordering among the terms appearing in a. We can now prove

Theorem 4.7.2 (Amitsur-Saltman [ 7 8 ] ) Let . ( E ,a. U, abelian crossed product in wh,ich r > 2 , n , = pf* > 1: 1 5 i non-degenerate. Then ( E ,a , U , u p ' ( ) is not a cyclic algebra.

be a generic U is

< r , and

Proof. This will follow by showing that if c E ( E ,a , U, u p ' [ ) satisfies cp E F ( [ 1 ; .. . ;(,) then c E F(E1,. . . Since ( E ,a , U ,a - l [ ) = E [ t ;a , UIc- it suffices to prove that if c f 0 E E [ t ;a , U ] satisfies cp E FIE1,.. . , E,] then c E F [ [ l , .. . :[,I. It is readily seen that if CP E F [ & ,. . . :<,I then A(?) = X(c)T. Hence if cP E F [ S l , .. . , E,] then X(c)P E F [ ( 1 : . . ,[,I. Suppose X(c) = c,,%trn, where ,m = ( m l, . . . m,) and t" = t y l . . . t,T.~.Thecondition X(c)p E F [ c 1 , .. . , <,I implies that 0"" = 1 for a" = 0;"' . . . cr,T.~,so either a" = 1 or rr" has order p. In the first case mi is divisible by n i , 1 5 i 5 r , which implies that X(c) E E [ ( I :. . . ,(,I. Since E ( E l ; . . . <,) is separable over F ( [ 1 . . . . ,(,), X(c)p E F [ [ l , .. . , E,] implies that X(c) E F[E1,.. . ,[,I. It remains t,o rule out the possibility that a' has order p. In this case, since r 2, there exists a aJ such that aTR@ ( a j ) and hence ( a 3 a") . is not cyclic. Let F" = Inv(a"). Then X(cjp = ( ~ e t " = ) ~N E / F , ( ~ m ) ( t 7 1 E L )Fp [ & ,. . . Hence tj.hTEIF,(c,)t"ptjl = NEIFm(cfi)tmp and since tJtmt;l = uJ,mtm(see P t 3 N E l F ,( ~ , ) t " ~ t j l= aJNEIE:-, (em)( 2 ~ ~ . ~ ~ t ~ ) ~

.I,[

.

>

1:

=a

j N ~ / (~~ % , ) N E / F ;(uj.mp)tmp. ~,

Hence

N ~ /( ~ i ~i L )=, lvE/Fe ~

( ~ ~ C ? % ) ~ ~(uj.7FL) E / F ,

and

~ ~ ~ ~ , ( ( a j c , ( u ~= , ~1.< ) Hence t,here exists a b E E such that (o"b)bP' =

or

184

IV. p-Algebras

contrary to the non-degeneracy of U.Thus we have that a" = 1 and X(c) E F [ [ l . .. . : t T ]Then . ( c X ( c ) ) pE F[[l,. .. and we can use induction on the number of non-zero terms in c to conclude that c - X(c) E .. and c = X(c) (c - X(c) E F [ < 1 , .. . , [ , I .

I,[,

+

I,[,

This and Lemma 4.7.1 irnply the existence of non-cyclic p-algebras that are division algebras. In fact, if we take into account the example preceding Lemma 4.7.1 and choose r = 2. n, = p in the generic abelian crossed product we obtain

Theorem 4.7.3.Let Fo be a finite field of characteristic p. T h e n there exists a ,non-cyclic p-division algebra of degree p2 over F = F0([1,( 2 , &), Ez indeterminutes.

V. Simple Algebras with Involution

The study of algebras with involution and especially the simple ones first arose in Albert's work on the multiplicatiorl algebras of Riemann matrices. This led llim to the syst,ematic study of algebras with involution for their own sake. His results were published in Structure of Algebras ([39]).A pa,rt of this chapter is devoted to presenting these results and recent extensions of them. We begin our discussion with a concrete determination in Section 5.1 of the simple algebras with involution. We base this on Wedderburn's description of the simplc algebras as algebras ErldDV where V is a finite dimensional vector space over a division algebra D and the isomorphism theorem that any isornorphism of Endn, Vlont,o EndD, Vz is induced by a semilinear isonlorphisnl of Vl/D1 ont,o Vz/Da In Sect,ion 5.2 we give criteria for a simple algebra to be involutorial, t,hat is, to have an involution. The formulation of the criterion for involutions of second kind in t,erms of corestrictions (see Section 3.13) is due to C. Riehm ([70]). The problem of determining the structure of involutorial simple algebras was considered in Albert's book, in which he solved the problem for central division algebras of degrees 5 4 and for division algebras wit21 irlvolutiorls of second kind of degree 2 over their centers. A cent,ral question of the structure theory raised by Albert is: Are involutorial central division algebras necessarily tensor products of qua.ternion algebras? In 1979 Arnitsur, Rowen and Tignol ([79])provided a negative answer to this question by constructing involutorial central division algebras of degree 8 that are not tensor products of quaternion algebras. On the other hand. Rowen has proved ( [ 7 8 ] )that any involutorial central division algebra of degree 8 is a crossed product (E.G , k) with the Galois group G that is a direct product of three cyclic groups of order two and Tignol has proved for char # 2 that for sucll an algebra A: n/12(A) is a tensor product of quaternion algebras. The Rowen and Tignol theorems are proved in Section 5.6 and the Amitsur-Rowen-Tignol counterexarliple is given in Section 5.7. This is based on the construction due to Arnitsur and Saltman of generic abelian crossed products which we gave in Section 4.6.

<

Any associative algebra with involution (A, J ) defines a Lie algebra K ( A . J ) of J-skew elements ( J a = -a) and a Jordan algebra H ( A , J) of J-symmetric elements ( J a = a). If the characteristic is # 2, the Jordan structure is given by the product a . b = 1/2(ab ba). On the other hand, if char A = 2 then it is necessary to adopt the quadratic Jordan point of view

+

N. Jacobson, Finite-Dimensional Division Algebras over Fields , © Springer-Verlag Berlin Heidelberg 1996, Corrected 2nd printing 2010

186

V. Simple Algebras with Involution

due to McC~immonbased on the composition (a, b) --t aba The relation between A and H ( A . J ) is especially close. For example. H ( A , J ) contains 1 and is closed under the power maps a --i o n , n = 0 . 1 , . . .. As a consequence one can extcnd the theory of the reduced norm to the Jordan algebras H ( A , J ) . This is developed in Sections 5.3, 5.5 and 5.6. Some of these results play a role in the structure theory of central division algebras with involution. e.g. in the proof of Rowen's crossed product theorem. An interesting question is to what extent the reduced norm of H ( A , J ) determines A. We consider this in Section 5.12 where aniong other ~ e s u l t swe show that if A and B are central simple algebras (not necessarily involutorial) then A and B are norm similar if and only if they are either isomorphic or anti-isomorphic. The proofs of the results on norm similarities make use on the one hand of the differential calculus of rational maps of vector spaces considered in Section 5.4 and on the other hand of the determination of the special universal envelopes of the algebras H ( A . J ) for (A, J ) central simple. The concept of special universal envelope is the Jordan analogue of the universal enveloping algebra of a Lie algebra. We consider this in Section 5.11. In Section 5.13 we consider the Jordan algebra H ( A , J ) . (A, J ) simple with involution, and we extend to H(A. J ) the results of 3.12 elements of rank 1.

5 .l. Generalities. Simple Algebras with Involution We recall that an involution J in an algebra A I F is an anti-automorphism of A such that J2= 1. Thus the defining conditions are:

+ b) = J a + J b ,

J ( a a ) = a ( J a ) , a: E F J(ab) = ( J b ) ( J a ) , J ( J a ) = a.

J(a

(5.1.1)

By an algebra wzth znvolutzon we sliall mean a pair (A, J ) where A is an algebra and J is an involution in A. We define a hon~omorphzsm of (A. J ) into an algebra with involution (B. K ) to be a map q of A into B that is an algebra homomorphism such that

or K ( 7 a ) = q ( J a ) for a E A. The class of algebras with involution over F constitutes a category with morphisms the homornorphisrns of algebras with involution. If q is a homomorphism of (A, J ) into ( B , K ) then ker 77 is an zdeal of (A, J ) in the sense that it is an ideal of A stabilized by J . Conversely, if I is an ideal of (A, J ) then (A, J) where A = A I I and 1is a + I --, J a + I is an algebra with involution and a --t ii = a I is a homomorphism of (A. J ) onto

(A, J).

+

We define the center C(A. J ) of (A, J ) to be C(A) fi H ( A , J ) where C(A) is the center of A and H(A, J ) = {h E A / J h = h } , the set of symmetric elements of A (under J ) . The algebra with involution (A, J ) is central if C(A, J ) = F (= F1).

5.1. Generalities. Simple Algebras with Involution

187

SVe shall now determine t,healgebras w i t h involut,ion ( A ,J ) that are simple i n t h e sense that A # 0 and A and 0 are t h e only ideals o f ( A ,J ) . Suppose first t h a t A is not simple and let B # 0 , # A b e a n ideal o f A . T h e n B n J B and B + J B are ideals o f ( A ,J ) . Hence B n J B = 0 and B + J B = A, t h a t is, A = B 8 J B , B and J B ideals. hloreover, B is simple since if D is a n ideal # 0 , # B o f B t h e n D + J D is an ideal # 0 , # A o f ( A ,J ) . Conrrersely. i f B is a simple algebra t h e n A = B $ B 0 wit,h t h e exch,ange involution E : ( b l , h a ) --t ( b 2 ,b l ) is a simple algebra w i t h involution. Now let A b e simple. T h e n A = E n d D V where V is a finite dimensional vector spac,e over t h e division algebra D . Let V * = H o m o ( V , D ) so V * is a right vector space over D w i t h [V*: Dl, = [V : Dl. Th'e can also regard V' as left vector space over Do b y defining dx* = x * d , d E D o = D , x' E V * . IVe write ( x ,y * ) for y X ( x ) ,x E V , y" E V * .T h e n (,) is a bilinear form from V x V * t,o D : (XI x a , ~ *= ) ( ~ I , Y * ) (xa,y*) ( x ,Y: + Y ; ) = ( x ,Y ; ) + ( x ,Y ; ) (5.1.3) ( a x ,y * ) = a ( x , y * ) , a E D (x,a y * ) = ( x , y*)a.

+

+

Also (, ) is non-degenerate i n t h e sense that ( x ,z * ) = 0 for all x implies z* = 0 and ( z , x X = ) 0 for all x* implies z = 0. I f e E A = E n d D V t h e n we have t h e transpose ex (or tl)E A* = EndDoVX such that ( e x ,y * ) = ( x ; ! * y * ) ,x E V, y* E V * .

(5.1.4)

T h e m a p !--i !* is a n anti-isomorphism o f A onto A*. Now suppose t h a t A has an anti-automorphism J . T h e n t h e m a p J a --i a* is a n isomorphism o f A onto A*. Hence there exists a bijective j-semilinear m a p s o f V onto V * . for an isomorphism j o f D onto D o , such that

Now j is a n isomorphism o f D onto D o . Hence, regarded as a m a p o f D onto itself. j is a n anti-automorphism o f D . Now put

T h e n g is a j-sesyuilinear form o n V / D i n t h e sense that g ( x . y ) is additive i n x and i n y and for a E D

Moreover, since ( , ) is a non-degenerate bilinear form from V x V * into D , g is non-degenerate i n t h e sense that g ( z , V ) = 0 + z = 0 and g ( V , z ) = 0 + z = 0. For .t E A a n d x , y E V , we h a v e g ( x , ( J 1 ) y )= ( x , s ( J t ) y )= ( x , e * s y ) = (!x, s y ) = g(!x, y ) . Hence J e is t h e uniquely determined adjoint o f e relative t,o g : (5.1.8) g ( e x , ' ~= ) g ( x , ( J ~ ) Y )2,, y E V.

188

V. Simple Algebras with In~olution

The existence and uniqueness of the adjoint comes from the fact that for a fixed y the map x --, g ( ! x , y ) is a linear function on V . Also for a fixed x the map ~j -,j - l g ( x , y ) is linear. Hence there exists a unique 1.' E V such that

If we take L to be the map x

-,g ( x , u ) u then

we obtain

since

g ( g ( x , u ) v , Y ) = g ( x , u ) g ( v ,Y ) = g ( x , j - l g ( v , y ) u ) . We now strengthen the hypothesis to: J is an involution. Since for t : x -, g ( x , u ) ~ : .J l : 7~ Y, g ( y , u l ) u , the condition J 2 = 1 gives g ( x , u ' ) v l = g ( x , u ) v for all x . u,v . This implies that there exists a c E D such that v' = cu for all Li E V. Then. by (5.1.9), we have

for a non-zero d E D and all 2 , y E V . Then j 2 g ( y , x ) = d g ( y , x ) ( j d ) . Choosing y and x so that g ( y , x ) = 1 we obtain d ( j d ) = 1 or jd = d - l . We now distinguish two cases: I. d = -1. Then j g ( y , x) = - g ( x , y ) and j 2 g ( y , x ) = g ( y , x ) which implies that j 2 = 1 SO j is an involution and g is a skew-hermitian form relative to j . If j = 1 then D is a field and g is a non-degenerate skew-symmetric bilinear form. Now suppose j # 1 and choose c E D so that q = j c - c # 0. Then jq = -q and j' : a --i q P 1 ( j a ) q is an involution. Moreover, h = gq is a hermitian form relative to j'. It is clear also that J coincides with the adjoint map relative to this form. 11. d # -1. Put q = d+ 1;h ( x , y ) = g ( x , y)q. Then h is sesquilinear relative to j' : u -,q - ' ( j ~ )and ~

+

+

Since ( , ~ ' ~ ) = q -q ~P 1 ( j q ) and gd = d - l . ( j ' q ) q - l d = ( 1 d ) - ' ( 1 d - l ) d = 1. Heiice j f h ( y , x) = h ( z , y ) which implies that g' is an involution and h is hermitian relative to 3'. Again J coincides with the adjoint map relative to h . We have proved

Theorem 5.1.12. Let A = EndDV where V is a finite dinzensionnl vector space over a d,ivision algebra D and suppose A h,as a n anti-automorphism J . T h e n D has a n anti-automorphism j and th,ere exists a non-degenerate j-sesquilinear f o r m g o n V / D such that J coincides with the adjoint m a p relative t o g. Moreove?; (f J i s a n involution th,en we m a y suppose that j is an, involution and either g i s herm,itian relative t o j o r D is a field, j = lo and g is skew symmetric.

5.1. Generalities. Simple Algebras with Involution

189

(We remark that, as is readily seen, we may equally well assume that g is anti-hermitian. It is sometimes advantageous to have such a g.) Let (vl , . . . , v,) be a base for V over D and suppose g,i, = g(v,, vk). Then G = (gzk)is invertible and for u = Ca,ui; v = Cbaui,az: b, E D we have

If :, is an involution then g is hermitian if and only if G is hermitian matrix: 7gza = 93z and if D is a field and J = l o then g is skew symmetric if and only if G is skew: gL3= -g,,. If L E A and we write tv, = C3fz7v,, L = (L,, ). (JL)v, = Zm,, (5,A 1 = (m2,)then the relations g(!v,, clc)= g(uz,( J k ) u k ) give 12,g,k = CEg,! ( j r n k e ) . Hence we have LG = G (j114) and

x3

The map L --, L is an ant,i-isomorphismof EndDV onto A'!ln (D) and L --i YjL) is an anti-automorphism. Hence q : 2! --, t ( j L ) is an isomorphism of EndDV onto M,,(D). The inap K : L --, G-l(tj-lL)G is an anti-automorphism of llTn(D). We have vJL = Y j M ) and KqL = GP1LG. Hence, by (5.1.14), 7J = K q so 7 is an isomorphism of (A, ,J) onto (hfn(D),K ) . Now suppose J, and hence K , is an involution. Then we may a,ssume j is an involution. We write a = j a and C* = JL. Suppose g is hermitian. Then it is known that except in the case in which D is a field of characteristic two and g is alternate in the sense that g(u, u) = 0 for all I L , there exists a base (vl.. . . ,u,) for V/D such that the matrix G = (g(vi,vj)) is diagonal (see Jacobson, [J3], v. 11, p. 153 and p. 171). The remaining case to be considered is that in which g is a.lterna.te since this is necessarily the case if g is skew and char D # 2. In the alternate case (including char D = 2) it is well known that n = 2n2 even and there exists a base (u1,7:1;u2,.. . ,urnurn)such that g(u,, yj) = 0 = g(v,,vJ), g(ui,v,) = hij = -g(zrj, w,) (loc. cit., p. 161). such a base is ca.lled symplectic whereas if G = (g(v,, v,)) is dia,gorlal then (vl, . . . , v,,) is called an orthogonal base. We have seen that if EndnV (or AJn(D)) has an involution then D has an involution. The converse is a,lso clear. Given an involution j : a --, Si in D we can define the hermitian form g such that g(Ca,ai, Cbjv,) = Caibj. Then this is non-degenerate and the adjoint map relative to g is an involution in EndDV. In matrix form j determines the invollition L --, L in hl,(D). Also if D is a field and [V : D ] = 2m then the adjoint map relative to the alternate form g ( x , a,ui b,vi, ciu,, divi) = Ca,d, - Cb,ci is an involution, the symplectic involution: in EndDV. In matrix terms this involution is

+ xt

x,,

+ xi

\Ire shall now sort out the central simple (A. J ) . Suppose first that the base field F is algebraically closed. If A is not simple we have A = B CE B0

190

V. Simple Algebras with Involution

and J = E is the exchange involution. Also B = M n ( F ) . Here the center of A is a direct sum of two copies of F and hence is two dimensional. Now suppose A is simple. Then the foregoing analysis implies t,hat either ( A ,J ) is isoirlorphic to ( h f , ( F ) , t ) where t is the transpose map or ( A .J ) is isomorphic to ( h & ( F ) ,t s ) where n is even and t s is the symplectic involution L --t S ( t ~ ) S - l as in (5.1.15). Evidently we have [H((lW,(F) 8 &f,(F), E ) : F ] = n 2 and [H((*Vl7, ( F ) .t ) : F ] = n ( n + 1 ) / 2 (where H ( A , J ) is the space of symmetric elements of A ) . The condition S ( t L ) S - l = L is equivalent to t ( L S ) = - L S . Hence [ H ( h f 7(,F ) ,t s ) : F ] = n ( n - 1 ) / 2 if char F # 2 and = n ( n 1 ) / 2 if char F = 2. In any case (&In( F ) ,t ) (iZ.I,(F),t s ) if n > 1. Since an isomorphism of an algebra with involution ( A ,J ) into a second one ( B .K ) induces a bijective linear map of H ( A , J ) into H ( B , K ) this is clear from the inequality of the dimensionalities of the H ' s if char F # 2. Now suppose char F = 2. We observe first that for any characteristic a matrix S is alternate if and only if S = T - t T . Hence if S is alternate then so is any cogredient matrix G S ( t G ) . Now if ( M , ( F ) , t ) E ( M 7 , ( F )t,s ) then there is an automorphism q of h f , ( F ) such that t = rltsq-'. Since 77 has the form L --, GLG-I this implies that G S ( t G ) = a l , a: E F * . Since a:l is not alternate this is impossible. Now let F be arbitrary, ( A ,J ) a central simple algebra wit,h involution over F and let F be the algebraic closure of F. If A is not simple then ( A ,J ) (Be B O ,E ) where B is simple and E is the exchange involution. Since C ( A ,J ) = F it is clear that B is central simple and hence B E = M,(F) for some n . If E is any extension field of the base field F of an algebra with involution ( A ,J ) then the linear extension J of J to AE is an involution in AE so we have the algebra with involution ( A E ,.J). It is clear that H ( A E ,J ) = H ( A , J ) E and hence [H( A .J ) : F ] = [ H ( A E J, ) : El. In the case in which ( A ,J ) 2 ( B 8 B ' , E ) and B is central simple we have ( A . J ) F ( h f n ( F )@ M,(F),E) and [ H ( A ;J ) : F ] = n2. Now suppose A is simple. Then the center C = C ( A ) is a field and J stabilizes C so either C = F or C is a separable quadratic extension of F and j = J / C is the automorphism # l c of C / F . If C = F, A is central ) F the algebraic closure of F . Then either simple over F and A F = M ~ ( Ffor (A,, J ) (&(F), t ) or (A,, J ) 2 (Adn( F ) ,t s ) . In the first case we say that ( A ,J ) is of orthogonal type and in the second ( A ,J ) is of symplectic type. If ( A .J ) is of orthogonal t,ype t,hen [ H ( A ,J ) : F ] = n ( n + 1 ) / 2 for A of degree n . If ( A ,J ) is of symplectic type [ H ( A J, ) : F ] = n ( n - 1 ) / 2 if char F # 2 and = n ( n + 1 ) / 2 for char F = 2, n the degree of A. If C ( A ) is a separable quadratic extension of F then A is a separable algebra so A p is semi-simple. Since C p = Fel @Fen where the ei are orthogonal idempotents exchanged by .J it follows that ( A p ,J ) r ((A&(F)B M ~ ( F E)) ,. Then [ H ( A ,J ) : F ] = n2.In this case J is said to be of unitary type. Also in the cases in which C = F, J is said to be of first kind while in the cases in which C # F , J is of second

+

"

"

"

kind.

For any ( A ,J ) we denote the space of skew elements ( J s = - s ) by K ( A , J ) . If char F # 2 then H ( A , J ) n K ( A , J ) = 0 and for any a we have a = $ ( a J a ) + ( a - J n ) and a Ja E H ( A , J ) , a - Ja E K ( A , J ) . Thus

+

+

+

5.1. Generalities. Simple Algebras with Involution

191

A = H ( A . J ) 33 K ( A , J ) i f char F # 2. O n the other hand, i f char F = 2 then H ( A , ,J) = K ( A , J ) # A unless A is commutative. I f char F # 2 and the center o f A is a quadratic field extension o f F and J = J I C # l c then C = F ( 0 ) where 8 = 30 = -0 and K ( A . J ) = 0 H ( A .J ) . Hence A = C H ( A .J ) C @ F H ( A , J ) . This last result holds also i f char F 2 and we can give a proof valid for all characteristics o f

"

Theorem 5.1.16. Let A I F have center C a quadratic extension of F and let J be an involution in A such that j = J I C # l c . Then A = C H ( A ,J ) 2 C %F H ( A ,J ) .

Proof. Since we have an automorphism # l c in C I F , C I F is separable. Hence there exists a 0 E C I F such that T K I F ( 0 )# 0. Replacing 0 by T c I F ( 0 ) - ' 0 we may assume that 0 + = T C I F ( Q= ) 1. T h e n the minimum polynomial o f 0 over F is X 2 - + p where p = N K I F( 0 ) and 4D + 1 # 0. I f a E A then we can verify that h l = ( J a - a ) ( l - 20)-' E H ( A , J ) . (5.1.17) Also h2 = a

-

Ohl E H ( A , J ) .

(5.1.18)

+

Hence a = hz Ohl so A = C H ( A ,J ) . Since H ( A , J ) n 0 H ( A ,J ) = 0 we have A C @ FH ( A , J ) . Again let A I F be central simple and let J and J' be involutions in A / F ( o f first kind). Since J'J is an automorphism i n A I F there exists an invertible b E A such that J' = i b J , that is, J'x = b ( J x ) b - ' , x E A . T h e condition J r 2 = l A gives b(Jb)-'z(Jb)b-' = x , x E A . T h e n (Jb)b-I E C = F 1 and Jb = pb, P E F * . T h e n b = J2b = P(Jb) = P2b so B2 = 1, B = k 1 . Thus either b E H ( A , J ) or b E K ( A , J ) . Conversely, i f b is an invertible element o f H ( A . J ) U K ( A , J ) t,hen i b J is an involution in A / F . I f b E H ( A , J ) t,hen z E H(A,ibJ) b(Jx)b-I = z # xb = b ( J z ) = J ( z b ) . Hence x -- xb is a bijective linear map o f H ( A ,i b J ) onto H ( A , J ) and hence [ H ( A ,J ) : F ] = [ H ( A ,i b J ) : F ] . I f char F # 2 this implies that i f h E H ( A , J ) then J and i b J are o f the same type. Similarly, i f char F # 2 and b is an invertible element o f K ( A , J ) then J and i b J are involutions o f opposite types. Now suppose char F = 2. T h e n H ( A , J ) = K ( A , J ) > ( 1 + J ) A . I f b is an invertible element o f ( 1 J ) A then i b J is o f symplectic type. For, i f F is the a.lgebraic closure o f F then J in A F = h f n ( F ) has the form L --, G ( t L ) G - l where t G = G and i b is L * (&I G ( t M ) G - l ) L ( M G ( t M ) G - l ) - l . T h e n i b J is L --, ( M G f t ( M G ) ) t L ( M G+t ( M G ) ) - I . Since M G f t ( M G ) is an alternate matrix it follows that i b J is o f symplectic type no matter what is the type o f J . W e note next that there exist invertible elements in ( I - J )A i f A has even degree. This is readily seen i f A = M?,(F). Hence it holds i f F is finite, so we may a,ssume F is infinite. Let ( u i ) be a base for (1 - J ) A over F . T h e n there exist 6, in the algebraic closure F o f F such that CGiui is invertible

+

+

+

V. Simple Algebras with Involution

192

and hence n ( C & , u i ) # 0 for the reduced norm n in AF = M n ( F ) . Since F is infinite t,here exist a, E F such that ~ ( C ~ , I#L0, )and hence Ccu,?~, is a n invertible element of (1 - J ) A . Now (1 - J ) A = K ( A . J ) if char F # 2 and = (1 + J ) A if char F = 2. The existence of invertible elements in (1 - J ) A for A of even degree and the foregoing results give the following theorem that summarizes these results. Theorem 5.1.19. Let A be central simple with involutio~zJ . Then: ( i ) A n y z n ~ o l u t i o ni n A has the form i b J where b is a n invertible element i n H(A: J ) U K(A, J ) . (ii) If char F # 2 and deg A is even then there exist in,uolutions of both types irr. A. Moreover: if J is of orthogonal (syrnplectic) type then the set { i b J / b t H ( A , J ) ,b invertible) is the set of orthogonal (symplectic) type involutions irt. A and { i b J 1 b E K ( A , J ) : b invertible) is the set of symplectic (orthogonal) type involutions i n A. (zii) If char F = 2 and degA is efuen then there exist invertible elements i n (1 J ) A and if b is such a n elemen,t then i b J is of symplectic type.

+

Now suppose A simple with center C separable quadratic over F and let J and J' be involutions of second kind in A I F . As for involutions of first kind. J' = i b J where J b = pb, P E K . Then b = J 2 b = J(,6b) = P ( J ~ = ) B,6b where 3 = ,ID. Hence AJcIF(P) = 1 so j3 = :tqpl. 7 E K . If we replace b by yb we obtain that b E H ( A , J ) . Conversely, if J is any involution of second kind in A then so is i b J for any invertible b E H ( A , J).

If .J, is an xiti-automorphisrn (involution) in A,, i = 1 , 2 , then the linear transformation J18J2such that ul'gaz --, J l u l 8J 2 a z is an anti-automorphism (involution) in A1 @ F A2. Now suppose the A, are central simple and the Ji are involutions. Then Al 8~ A2 is central sin~pleand J1@ J2is of ~ r t ~ h o g o n a l type if J1 and Jz are of the same type and char F # 2. Otherwise, J18 J2 is syrnplectic. It suffices to prove this in the algebraically closed ca,se. Then it follows from t h e readily verified fact that the tensor product of the unit rrmtrix (of any size) with a "matrix S ' o f (5.1.15) is a n alternate matrix and the tensor product of t ~ v omatrices S is symmetric or alternate according as char F # 2 or char F = 2. Now let (D.j ) be an algebra with involution j : d -- 2 and consider t h e rnatrix algebra 12ir,(D) ~vhichwe can regard as h f 7 , ( F ) D. Then we have the involution t ?t j in n/l,,(D) which is called the standard involution (based on j ) . This has the form (dill *t (&). (5.1.20) If g,, 1 5 i

5 n,, is a n invertible element of H ( D ; j ) then the matrix G = diag{g,. . . . , g,)

(5.1.21)

is a syrninetric matrix relative t o the standard involution (5.1.20). Hence

5.2. Existence of Involutions in Simple .Algebras

193

is an involution in Al,(D). We shall call this a canonzcal znvolutzon based on j and G. Using these definitions we can reformulate our results on the determination up to isomorpl~ismof the algebras with invol~ition(A. J) for which A is simple in the following matrix form Theorem 5.1.23. Let (A, J) he a n algebra with involution such that A is simple. Th,en either there exists a division algebra with invol7~tion( D ,j ) and a n integer n such that (A, J ) is isomorphic to a pair hfn(D) with a canonical ir~volution,or there exists a field D I F and a n even integer n = 2m such that (A: J ) (11/1,(D): t s ) .

"

We note also that the second case can also be regarded as a matrix algebra with standard involution. For this purpose we regard ,$fn (D) as 1\.f,,(Af2(D)). We take j in B = &f2(D) to be

Then it is readily verified that the involution in Afn(D) defined by (5.1.24) coincides with the standard involution in Mm(B) based on 3 .

5.2. Existence of Involutions in Simple Algebras We shall obtain conditions for a simple algebra to be involutorial. that is. have an involution. We consider first the case in which A is central simple so A = lVT(D) where D is a central division algebra. Note that A has an anti-auton~orphismu [A] E B r 2 ( F ) the subgroup of Br(F) of [A] such that [A]' = I . This follows from the following facts: [A]-' = [A0].A has an antiautomorpllisrn % A A0 and since A and AO have the same degree A r AO % -4 A'. In particular, if A has an involution then A @ p A -- 1. A remarkable fact discovered by Albert (see [39]. p. 161) is the converse:

-

"

Theorem 5.2.1. If A @ F A

-

1 then A has a n znvolution.

The proof w e shall give of this result is an unpublished one due to Tamagawa that is constructive. This is based on the observation that for any A there is a unique automorphism .ir of A @ F A such that

Also, if B is the subalgebra of T-fixed elements then B is spanned by all a 8 a , a E A a n d i f (u1, . . . . u,) isabase for Athen (u,@u,,u,@u,+u,&u,,i < 3 ) is a base for B. Hence [B : F] = n(n 1)/2. If A is central simple then T is

+

194

V. Simple Algebras with Involution

inner so there exists a n invertible f E A 8 A determined up t o a scalar such that b @ a a f(a@b)fP1. (5.2.3) Then B is the centralizer of f in A @ A and f = y 1,7 E F" since 7r2 = 1 A ~ A 1%now have Lemma 5.2.4. If AAX A

1. t h ~ n7 IS a square zn F

-

Proof (Darrel Hazle). Otherwise. F[f]is a subfield of A BE.A. If A is central simple of degree m and A % F A 1 then by the centralizer theorem (BA 11, p. 224). [B : F] = [A 8~A : F]/[F[f]: F] = m4/2 contrary to [B : F] = m 2 ( m 2 1)/2.

+

We car1 now give the

-

Proof of 5.2 1. Let the notations be as in the foregoing discussion. Since we are assuming that A @rA 1 we may identify A @ A with EndFV where V is a n m2 dimensional vector space over F . Also since y is a square we may assume ,f2 = 1. Then. by the Jordan decomposition of the linear transformation f , we have a decomposition V = Vl 8 V2 @ . . . 33 V, where V , is either one or two dlmenrional and indecomposable under the action of F [ f ] . If char F # 2 then every V , = Fu, with fu, = &v,. If char F = 2 we have at least one such V,. Otherwise. all are tu70 dimensiorlal and all are isomorphic as F[f]modules (isonlorphic to F [ f ] ) . Then r = m2/2 and the centralizer of f in A @ A is isomorphic to the matrix algebra 1tfm2/2(F[f])whose dimensionality is m 7 2 again contradicting [B : F] = m 2 ( m 2+ 1)/2. Hence in all cases we have V = U @ V17 where [U: F] = 1 and U and IV are stabilized by f . Let e be the projection on U determined by this decomposition. Then e E B and e is a one idempotent of EndFV = A g A. Then (A 8 A)e is minimal left ideal of A A so [(A 8 A)e : F] = [V : F] = m2. At this point we assume, as we may. that A is a division algebra. So (L + ( a @ 1)e is injective and [(A @ 1)e : F] = [A : F] = m2.Then Then for any a E A we have a unique a* E A such that

Evidently ,J : a

+

a* is linear. Moreover,

(1 8 (ab)*)e = (ab 8 1)e = (a 8 l ) ( b 8 1)e = (a &? 1)(1 8 b*)e = (1 @ b*)(a @ 1)e = (1 8 b * ) ( l @a*)e = (1 8 b*a*)e Hence (ab)* = b*a*. Finally, 1 a*)e) = (a* 8 l ) e (1 8 a ) e = ~ ( ( @a 1)e) = ~ ( ( 8 = ( 1 8 a**)e so a"* = a . Hence J is an involution.

.

5.2. Existence of Involutions in Siinple Algebras

195

We consider next the case in which the center of A is a separable quadratic rxterlsiorl C of F and we seek a condition that A has an involution J such that J 1 C is the automorphism # 1 in C . The result, more or less due to Albert [39]. p. 159. can be stated in modern terminology as follows (see Riehm [70] and Tits [ 7 1 ] ) .

Theorem 5.2.7. Let A be simple 102th cen,ter n separable quadratic extens i o n C I F . T h e n there exists a n involution J i n A I F such that J / C i s t h e a u t o m o r p h i s m j # 1 o f C I F if and only if corclFA -- 1 . The proof we shall give is due to JV. Scharlau [85].It is based on the study of the simple algebras over C that possess anti-automorphisms .J such that J C = 2 . As for involutions, such an anti-automorphism of A / F is said to be of second kznd. Let J be such a map. We write J a = a " , a E A, and ga: = 6.0 E C. Since J 2 / C = j 2 = l C and since J 2 is an automorphism there exists an invertible b E A such that

where b is determined up to a multiplier in C . Now consider J 3 . Since J 3 =

JJ2 = J2J (bxb-')" = (b"jK1x*b" = bx*b-l. Hence b*bx" = x*b*b, x E A, so b*b E C . Since (b*b)* = b X b ,b*b = 0 1 , P E F . Also b(b*bjb-I = b ( p l ) b K 1= ,!?l so bb* = b*b. Hence

bb* = $61= b*b,P E F * . Replacement of b by yb, y E F * , replaces ;;iyb*b = N c l F ( y ) P . We now have

P

(5.2.9)

by P N c I ~ ( y )since ( y b ) * y b =

Lemma 5.2.10. Let A be a diriision algebra over F with center a separable quadratic extension C of F and suppose A has a n anti-automorphism J : x --, 2" of second kind. T h e n J 2 has th,e f o r m x --, bxb-', b E A, and (5.2.9) holds. Aloreover, if if N C I F ( K wt )h e n A has a n involution of second kind.

Proof. If p = N C I F ( y ) then by replacing b by y K 1 b we may assume /3 = 1 so bb* = 1 = bXb.If b = -1 then clearly J itself is an involutiori. Hence assume that b # -1. Then b + 1 is invertible in A and direct verification shows that

is an involution. Since its restriction to C is the same as that of J this is of second kind. Again let A be any sirnple algebra over F with center C . As in Section 3.13, we can define the algebra A which is identical with A as ring but in which the action of C is twisted by 3 . If J is an anti-automorphism of second kind in A then J is an isomorphism of A onto A0 whose restriction to C is j . Then

196

V. Simple Algebras with Involution

-

-

-

7. is a n isomorphism of A / C onto A O / C . Since A @cAO 1 it follows that ~ g ~ 1. i The i converse follows since A E ~ A 1 implies that A/C S AO/C. Thus A has a n anti-automorphism of second kind if a.nd only if A E c A -- 1. It, is clear that if AlmCA1 1 and A 2 s c A z 1 then ( A ~ x ~ A ~ ) % A2= ~ A ~ (Al 3~A2) 8~(A1 XC A2) 1 and A'f,,,(K) @ M m ( K ) 1. Hence me obtain a subgroup r of Br(C) consisting of the [A] such t,hat A ~@c A 1. For any central simple A / C we have the j-semilinear automorphism 3 of A A gc such that a @ b -+ b @ a . The F-subalgebra of A of fixed points is the corestriction corclFA (Section 3.13). We have seen that if ( c l ; . . . ; e n ) is a base for A I C then every element of corclFA has the form C c u , 3 e , e 3 where a,, = 6,j E C (p. 151). If z # j , a e , % e ; , + 6 e , B e , = cue, 8 e, e, g a e , = (aei e,) g (cue, ej)-ae, 8 a e , - e j @ ej. It follows that rorclFA is F-spanned by the element,^ a @ a: a E A. We recall also that cor A = A E ( c ~ r ~ , ~ cAo )r c~, ~ . A is central simple over F. and we have the corestriction homorriorphisrn [A] --i [corciF A] of B r ( K ) into B r ( F ) . Restricting t o r we obtain a homomorphism into a subgroup of B r ( K / F ) represented by quaternion algebras. More precisely we have

-

--

-

- -

3-

+

+

+

Theorem 5.2.12 (Scharlau [ 8 5 ] ) . Let [A] E T and let J be a n antia'u,tom,orph,ism of second kind i n A, b a n elem,en,t of A s u ~ hthat J2 : x * bxbpl, and b(Jb) = 6 E F . T h e n c o r c , ~A

- (C,

j : 0)

(5.2.13)

Proof. \tTe have a n action of A a K A on A such that ( a 2 b)s = axbX(h*= Jb). Restrictirlg t o corKIFA we obtain a n action of corKIFA on A / F in which (n%a).r = a r n * . Since B = corc,j- A is simple we can identify B with its irnage in ( ( ~ n d ~which ~ ) is ~ a) quaternion algebra. Since B = ( ( ~ n d ~ A ) ~B) ' . ( E ~ I ~ ~ Now7 A ) if~ cu . E K , a . x E A then ( a @ ::)ax = a ( a z ) a * = ~ ( a z a * ) . Hence a E ( E n d ~ ~ 4Next ) ~ . consider the map u : x --i b".r" where b is a n invertible elerrlent of A such that .c** = bxb-l. Then

-

Thus u ( a x a X )= a ( u ( x ) ) a * so

2~

E (EndFA)B. Moreover,

Hence u2 = 131 and since u ( a z ) = b a ( a z ) * = bb*z* = a u ( z ) ; we have the relations 'UCY = b u , u2 = @ (5.2.14) in C + C u . It follows that C helice B (C, j , 0).

-

+ Cu " (C:j , 0 ) .Then C + C u = (EndFA)

and

5.3. Reduced Norms of Special Jordan Algebras

197

-

-

J e can now give the proof of Theorem 5.2.7. Suppose B = corclFA 1. Then A $ 3 2 ~ -. Bc 1 so we have the hypothesis of Theorem 5.2.12. Hence, by 5.2.12. (K,j,,O) -- 1. Then 3 E -VKIF(Kx). Since we may replace A by any similar algebra, we may assume A is a division algebra. Tlien, by 5.2.10, A has an irlvolution of second kind. Conversely, suppose A has an involution J of second kind. Then the 3 determined before is 1 and ( C , j ,0) 1. Hence c0rclFA 1. 0

-

-

-

Remark. Since ( K ,y,P1) (I<.J. 02) if and only if PI and B2 are in the sairie roset rriod NKIF(Ks) it follows from 5.2.12 that if J1 and J2 are two anti-a~itomorphismsof second kind in A and 01 and p2 are the corresponding elements of F" then Dl = pz (mod iVI
-

5.3. Reduced Norms of Special Jordan Algebras If (A. J) is an algebra with involution then we have seen that the set H (A. .T) of symmetric elements of A under J is a subspace. Also 1 E H ( A , J ) . It is natural to seek to endow H(A. J ) with a structure that is richer than that of a vector space with unit. If char F # 2 the important additional structure. first suggested by the physicist Pascual Jordan around 1932, is that given by the cornposition a . b = +(ab+ ba). Using this. the unit element 1 acts as unit: 1 . a = a = a . 1. If char F = 2 the composition a . b is not available and though a o 11 = a11 + ba is, this has the serious drawback that n o 1 = 0 for all a. It has t u r i l ~ dout tliat the right composition to use for H(A. J ) in all characteristics is ( a , b) --i U,b = aba which is linear in b and quadratic in a. We recall that me have already encountered this in defining the variety of clemeilts of rank one in an algebra (Section 3.12). There is a well developed abstract theory of Jordan algebras based on the composition U,b (see Jacobson [69] and [81]). For the applications we have ill mind we shall not require this; it will suffice to consider the concept of a special Jordan algebra defined as follows. Definition 5.3.1. If A is an associative algebra; a special Jordan algebra (in A) is a subspace H of A such that 1 E H and Uab E H for a, b E H . A is called the asmbient algebra of H . As in Section 3.12, we define bilinearization U,,b of Ua by Ua b = Ua+b U, - Ub. Then U, bc = acb bca E H for any a , b, c E H . If H is a subspace of A containing 1 with base ( u l . .. . , u n ) then H is a special Jordan algebra in k for 1 5 2 , j. k 5 n. These conditions A if and only if U,,vk and U u a , U SE~ H are evidently necessary. On the other hand, if they hold and a = Ca,u,, b = C B k u k then

+

198

V. Simple Algebras with Involution

Evidently, these conditions on a snbspace imply that if H is a special Jordan algebra in A then HE is a special Jordan algebra in AE for any extension field

EIF.

-

-

There are a number of other important closure properties of special Jordan algebras. We note first that if H is a special Jordan algebra then [abc] [[ablc]= (ab - ba)c - c(ab - ba) = abc cba - bac - cab = U,.,b - Ub,,a E H if a , b, c E H . Also a 0 b ab ba = U,,bl and [abI2 = (ab - b ~ = )abab ~ baba - ab2a - ba2b = a o Uba - U,b2 - Uba2= b o Uab - U,b2 - Uba2E H if a , b E H . Another important closure property is that an E H ; n 2 0 if a E H. This follows by induction since ao = 1 E H , a E H and anL2 = U,an E H . In addition to the operators U, and Ua,b it is convenient to introduce the operator Va,b defined by V,,bx = U,,,b and V, = U,,l. In the ambient associative algebra we have the following dictionary:

+

+

U,x = axa U,,bx = axb Va,bx= abx

+

+ bxa + xba

The abstract theory is based on identities connecting these operators. Of particular importance is the so-called "fundamental formula"

which holds since

U,&U,x

= a(b(axa)b)a

UUnbx = (aba)x(aba) which are equal by the associative law in A. If char F # 2 we have a . b = $ a o b and, as is readily verified,

Hence if char F # 2 then a subspace H of A is a special Jordan algebra if and onlyiflEHanda,b~Hforanya.b~H.

Examples 1. For any associative A we have the special Jordan algebra H = A (in which the composition ab is replaced by aba). we denote this as A+. The definition 5.3.1 amounts to: H is a special Jordan algebra if H is a subalgebra of A+ where a subalgebra is defined in the obvious way. 2. Let ( A ,J) be an associative algebra with involution. Then H ( A , J) is a special Jordan algebra. We observe that A+ can also be viewed as an algebra of

5.3. Reduced Norms of Special Jordan Algebras

199

symmetric elements of an algebra with involution. Let ( B = A @ A'. E ) where is the exchange involution (a. b) -, (b.a ) . Then N ( B . E ) = { ( a ,a ) / a E A) and the map a --i ( a , a ) is an isomorphism of A+ onto H ( B ,E ) where an isomorphism of H I onto H z is defined to be a bijective linear map a1 --, a2 such that 1 1 and U,,bl --, U,,bz for a1 -,a2, bl -,b2. 3. Another important class of examples of special Jordan algebras are special Jordan algebras defined by Clifford algebras. Let C ( V . Q ) be the Clifford algebra of a quadratic form Q on the vector space V (see e.g. BA 11. p. 228f). We recall that the canonical map of F 1$ V into C ( V ,Q ) is an isomorphisnl so we can identify F 1 @ V with its image in C ( V .Q ) . Then C ( V ,Q ) is generated by V (and 1) and we have the defining relations E

+

where Q ( v .w)= Q ( v w) - Q ( v )- Q(u1).Let a = a 1 + v , a E F,v E V . Then n2 = ( a 2+ Q(11))l+ 2av. Hence

where

T ( a ) = 2a, N ( a ) = a2 - Q ( v ) . By (5.3.5) we have for a = a 1

(5.3.6)

+ v , b = pl + w,

+

where AT(a.b) = N ( a b) - N ( a ) - N ( b ) . If we multiply (5.3.7) on the right by a and simplify using (5.3.5) we obtain

+

uba = ( T ( a ) T ( b ) N ( a ,b))a N ( a ) b - T ( b ) N ( a ) l .

(5.3.8)

Thus F l + V is a special Jordan algebra in C ( V % Q ) . We recall also that C ( V ,Q) has an involution J, called the mazn znvolutzon, such that Jv = v for all v E V. This defines the special Jordan algebra H ( C ( V ,Q ) ,J ) # F l + V if [ V :F ] > 1.

If H is a special Jordan algebra in A t,hen, as we saw above, ar E H for . . . ar where X is an indeterminate every a E H . If f ( A ) = aoXr alXr-I and the a, E F then we define f ( a ) = aoar a l a r - l . . . + a r l E H and we define the minimum polynomial p,(X) of a to be the monic polynomial E F[X]of least degree such that p,(a) = 0.This is the same as the minimum polynomial of a as element of A. Now let ( u l , . . . , u,) be a base for H / F arid let ( 6 1 % .. . , ),[ be indeterminates. Let x = C[iu, E H F ( < )F , ( [ ) = F(C1,.. . ,),[ so x is in the special Jordan algebra HF(<)in AF(0. As in Section 1.6, we consider the minimum polynomial of x in HF(<)which we denote as

+

+ + +

+

where T,(E) E F ( [ ) .If we supplement the base for H / F to a base ( u l , . . . , uN) for A I F and introduce additional indeterminates En+',. . . ,tN we obtain a

200

V. Simple -4lgebras with Involution

generic element C: c J u j of A whose minimum polynomial is contained in FIE1.. . . , E N . A] (Leinma 1.36). This is homogeneous in the ('s and A. If we put En+l = . . . = = 0 in this polynomial we obtain m,,A(x) = 0: which implies that m,.H (A) / n i , , , ~(A). Hence, by Gauss' lemma. r n , , , ~(A) E I?[(. A] and m,,II(X) is homogeneous of dcgree m in the $, and A. Accordingly. T ~ , ~ ~ ( $ ) in (5.3.9) is homogeneous of degree i in El, . . . , Em. \LTe call now carry over verbatim the discussion of the reduced charact,eristic polynomial, trace and norm of an associative algebra given on pp. 24-29. If a = Cy a i u i , a, E F , we define

cN

-

.

T , ~ J J (. .~. ~a,). , and we call this polynomial E F [ A ] the where 7% (a) reduced (or generzc) nlznzmum polynomial of a in H. This is indep~ndentof the choice of the base for H I F and is independent of base field extension: If E is an extension field of F and a E H ( c H E ) then n l , , ~ ( A ) = m, H,(A). The T%,Hare polynomial functions on H and we call

the reduced (or generzc) trace and reduced (or generzc) norm respectively of a E H . The integer m is called the degree of the special Jordan algebra H. As in the case of a n associative algebra (p. 25) it is readily seen that m, H ( a ) = 0. Hence pa(A) 1 m, H(A) for, p, (A) the ininiinum polynomial of a. Also the proof of Proposition 1.6.9 goes over to prove Proposition 5.3.12. If F is znfinite the subset R of a E H such that p,(A) = V Z , , ~ ( Ais) a non-vacuous Zariski open subset of H.

Let d(x) = d(E1,. . . .En) be the discriminant of m,,H(A) This is a hornogeneous polynonlial E F[E]. Mre shall call H unram~fiedif d(x) # 0. In this case the set U = {a E H / d(a) # 0) is non-vacuous Zariski open in H and the condition a E U is necessary and sufficient that ma J ~ ( A )has distinct roots in any algebraically closed field F > F. Suppose a E U n R. (We shall see in Section 5.5 that U c R.) then pa(A) has m distinct roots in F. Then in F [ a ] , which is a subalgebra of HF, we have m non-zero orthogonal idempotents e, with Ce, = 1. We note that the orthogonality conditions e,e:, = 0, z # j are equivalent to the Jordan conditions

In terrns of the associative product these are e,ej together with e: = ei imply eie, = 0, i # j .

+ ejei = 0, eieje, = 0 which

Lemma 5.3.14. If H contains r non-zero orthogonal idempotents ei with .Erei = 1 then r < deg H. Moreover; H is unramified if and only if HF for F a n algebraically closed field containing F , has m = deg H non-zero orthogon,al idempotents ei with Cei = 1.

5.3. Reduced Norms of Special Jordan Algebras

201

Proof. If H contains e, f 0 such that e: = e,, e,e, = 0 for L f J and C e , = 1 then we can choose r distinct a, in F . If a = Ccu,e, then p,(A) = nr(A- a,). Then r deg H p = deg H . If r = deg H then p,(A) = m,(X) and d ( a ) # 0. Then d ( r ) # 0 and H is unramified. Conversely, suppose H is unramified. Then in H E we can clloose a E R n 11. Then m,(X) = p,(X) llas m distinct roots and F [ a ]contains m non-zero orthogonal idempotents e, with C e , = 1.

<

We call now determine the reduced minimum polynomials of the special Jordan algebras H ( A , J ) for ( A .J ) central simple with involution. We consider first the case of an algebraically closed field F . Then the simple algebras with involution are: 1. (Al,,,(F) @ 11lm(F)O.E ) , 2. ( A f m ( F )t,) , 3. ( - t f m ( F )t,s ) . m even. 1. We have seen that H ( h f n , ( F )8 hl,(F)O) r lZfn,(F)+.Now it is clear from the definitions that if 7 is an isomorphism of a special Jordan algebra H with a special Jordan H2 then m,,p, ( A ) = m,,,H, ( A ) . Hence it suffices to consider the Jordan algebra M m ( F ) + .Also since M m ( F ) + and the as~ociat~ive algebra A I m ( F ) have the same power structure it suffices to determine the reduced cllaracteristic polynomial for the associative algebra AIT,(F). This we have done in Section 1.6 where we showed that m,(X) = x,(A) = det(A1 a ) . Also we showed that MT,(F) is unramified (p. 27) which implies that ( H ( f L f m ( F0) hf,,(F)O, E ) is unramified. 2. H ( h f m ( F ) t, ) . If z = C y Eiieii C,,, EZ3 ( e t j e j i ) where the and C3 are indeterminates then z is a zero of x,(A) = det(X1 - z ) . Hence, deg H ( A l m ( F ) ,t ) m. On the other hand, the diagonal idempotents e,, are symmetric matrices. Hence, by Lemma 5.3.14, H(_bfm( F ), t ) is unramified and has degree m and r n , , ~ ( X )= x,(X) for H = H (iWm( F ) ,t ) . 3. H(&I,(F), t s ) , m = 2r. As shown on p.192, we may regard Af7,(F) as M,(D) where D = IlJ2(F).Then t s is the standard involution in h/I,.(D) based on the involution d --i d = ( T r d ) l - d in D. Thus t s is

+

+

c3,

<

dl1

dl2

. . . dlr

4-1

. dr2

... ... . . . drr

-

dl1

6,

-

-

... d r ~

<21

... ...

&r

drr

and H(,VI,.(D), t s ) is the set of matrices

dl1 a=

dl2

.

. . . dlT .... ..

ZIT &,. . . .

. d,,

:

d , , = dti.

(5.3.16)

If char F # 2. the condition d,, = d,, holds if and onlv if d,, = & I 2 . 6, E F . In all characteristics. a E H ( M m ( D ) ,t s ) if and only if t ( a S ) = - a s . If char F # 2 it is proved in Jacobson. [fig], pp. 230-232 that deg H ( M m ( F ) ,t s ) = r = m / 2 . that this Jordan algebra is unramified

202

V. Simple Algebras with Involution

and m , , H I F ( X ) = Pf(SX - a s ) where Pf is the Pfaffian of the alternate matrix SX - a s . Now suppose char F = 2. Then the condition a,, = a,, = (tr r~,,)1- a,, is equivalent to t r a,, = 0. It can be shown that in this case m , H ( X ) = P f ( S X 2- a 2 S ) and H = H ( A . J ) is ramified (Jacobson [76]. p. 137). We shall not require this result. More interesting is the reduced minimum polynomial , M'e observe first that if ( A . J ) is any in a certain subalgebra of H ( - M m ( F )t,). associative algebra with involution then ( I + J ) A is an outer ideal in H ( A , J ) in the sense that this is a subspace of H ( A , J ) and U,b E (1 + J ) A for every b E ( 1 + J ) A and every a E H ( A . J ) . The verification is immediate. It is clear also that if char F # 2 then ( 1 J ) A = H ( A , J ) and that ( I J ) A is a subalgebra of H ( A , J ) if and only if 1 E ( I J ) A . That this is not always the case can be seen in the example in which A = iVfm(F),char F = 2. and J = t the transpose involution. Here the diagonal entries of any ( I + J ) a are all 0 so 1 @ ( 1 J ) A . Now consider the subspace ( 1 ts)LJ\/fm(F) of H ( h f n , ( F ) . t s ) . We claim that it consists of the matrices of the form (5.3.16) in which the d,, are scalar matrices: d,, = &,I2.To verify this let E,,, 1 z.3 5 r = m / 2 , be the matrix of the form of the left hand side of (5.3.15) with dZ3 = 12 and all other dkp = 0. Then any a E ( 1 t s ) M m ( F ) has the form given in (5.3.16) with d,, = &,I2, 6,E F , since d + d = (tr d)12. On the other hand. let a be any matrix of the form (5.3.16) in which d,, = &,I2. 6, E F . Then

+

+

+

+

+

<

+

a = (1

+ t s )b where b = CtCjdi3E i j and dii = ( -

:) . 1t is clear from this

that [ ( l + t s ) A f m ( F ): F ] = m ( m - 1 ) / 2 ( m = 2r). Moreover, it is easy to check that the argument using Pfaffians in the characteristic # 2 case carries over to show that for H' = ( I t s )M m ( F ) we have r n , , ~( A~) = Pf (SX - S a ) , H' is unramified and deg H' = m / 2 . It is clear by extension of the base field that if A is central simple over any F of cllaracteristic 2 with involution J of symplectic type then H 1 ( A .J ) = ( 1 J ) A is a subalgebra of H ( A . J ) and [ H 1 ( AJ. ) : F ] = m ( m - 1 ) / 2 . The results for algebraically closed F can be extended to the special .Jordan algebras H ( A . J ) and H 1 ( A ,J ) for central simple ( A .J ) over any field F as follows.

+

+

Theorem 5.3.17. 1. Let A I F be central simple of degree m . Then H ( A @ A', E ) ( 2A + ) is unramified of degree m ~ n d m ( ~ . ~ =~ xa(X) , ~ ( a=)det(X1-a) where a is regarded as an element of A F = M m ( F ) ; F an algebraically closed extension of F . 2. Let A be simple with center a separable quadratic extension K / F , J an inuolution of second kind in A . Then H = H ( A , J ) is unramified of degree m = deg A / K and m , , H ( A ) = m , , H I K ( A ) (as in 1.) for a E H . 3. Let A be central simple of degree m over F , J an involution of orthogonal type in A. Then H ( A , J ) is unramified of degree m and m a , H( A ) = m a , A ( A ) . 4. Let A be central simple of even degree m over F , J an involution of sgmplectic type in A . Then HI = H 1 ( A ,J ) = ( 1 J ) A is unramified of degree

+

5.3. Reduced Norms of Special Jordan Algebras

203

m / 2 and rn,,H, (A) = Pf(SA - a s ) where a is regarded as a n element of Ap =

Afm(F) such

that S ( t a ) S - l = a for S as i n (5.1.15).

Proof. 1. We have seen that a --, (a; a ) is a n isonlorphism of A+ onto H ( A 9 A", E ) and that A+ and A have the same power structure. Hence the result in this case follows from our discussion of the reduced characteristic polyrlomial of A given in Section 1.6. 2. We may assume F infinite, since by Wedderburn's theorern, if F is finite A = ikfv,(K) and the result is trivial in this case. We observe also that the Zariski topology for H as vector space over F is the restriction t o H of the Zariski topology for A as vector space over K since A = H K . Now let a E H ( A , J ) and let p,(X) be the minimum polynomial of a in A I K . Applying J to p,,(a) = 0 and we see that ,u,(X) E F[A]. Also the subset of H of the a such that p,(A) = m,,H(A) is non-vacuous Zariski open and the subset of a E A such that p,(A) = m,,HIK(A) is open and hence its intersection with H is non-vacuous and open (since A = H K ) . Hence on this open subset of H we have rn,~H(A) = ma,HIK(X).It follows that this equality holds for all a E H ( A , J ) . By 5.1.16, A = K H ( A , J ) H ( A , J ) K . Since the reduced rrlinimum polynomial is independent of extension of the base field, the fact that H ( A , J ) is unramified of degree rn follows from these properties for A+ and the associative algebra A. 3. This follows by a field extension argument from the result in the algebraically closed case. 4. The same is true for symplectic t,ype invollitions since in all cases H1(A,J), = H f ( A F ,J ) . An important consequence of Theorem 5.3.17 is the following theorem on commutative subalgebras of simple algebras with involution. Theorem 5.3.18. 1. Let A be a simple algebra with center a separable quadratic extension K I F , J a n involution of second kind i n A / F . T h e n A / K contains a commutative subalgebra E / K of the form K E F E o uihere Eo = F [ a ] , a a separable element of H(A, J ) such that [Eo: F] = deg(A/K). 2. Let A be a cen,tral simple algebra with involu~tionJ of orthogonal type. Tlten A contains a separable commutative subalgebra E such that [E : F ] = deg A and E is contained i n H ( A , J ) . 3. S a m e hypothesis as i n 2 with J of symplectic type. T h e n A contnins a separable commutative subalgebra EIF such that [E : F] = degA and E c H f ( A ,J ) .

Proof. We nlay assume F infinite. Put H' = H ( A , J ) in all cases except char F = 2. J of symplectic type in which case we put H' = ( I J ) A . Using a Zariski topology argument we see that there exists an a E H' such that pa(X) = ma,HI(A) and d ( a ) # 0. It is readily sern that in case 1, E =

+

204

V. Simple Algebras with Involution

K [ a ] Eo , = F [ a ] satisfy the stated conditions. In all the other cases E = F [ a ] satisfies the conditions.

5.4. Differential Calculus of Rational Maps In the sequel me shall require the differential calculus of rational maps of vector spaces. It is convenient at this point to record the basic fact,s of this calculus. Proofs can be found in a number of places, e.g. Jacobson, [68]. pp. 214-221, T.A. Springer, [73], pp. 1-8. Let V be an rn-dilnensional vector space over an infinite field F , P ( V ) and Q ( V ) the algebras of polynonlial functions and of rational functioiis respectively on V . P ( V ) S F [[I = F I E 1 , . . . , whcre m = [V : F ] . An isomorphism of these algebras is obtained by choosing a base ( u l . . . . ! u,,) for V . Then the map of f ( E l , . . . ,Ern) into the function f : u = C a , u i --, f ( a l , .. . .am,)is an isomorphism. If we use the usual identification of functions defined on Zariski open subsets of V , we can regard Q ( V ) as the field of fractions of P ( V ) . If E is an extension field of F then any f E P ( V ) has a unique extension to f E P ( V E ) and the same holds for any f E Q ( V ) . Any a E V det,ermines a unique derivation A" in Q ( V ) such that for any u" E V * (the vector space of linear functions on V) we have

I,<

A a u * = u* ( a ) .

(5.4.1)

For f E Q ( V ) and c E V at which Q is defined me write A: f = (A"f ) ( c ) and we call this the d~rectzonalderr11at7oeof f at c In the dzrect~orla . I f f is given by f ( [ I ,. . . [,) = g ( [ l . . . . . t r n ) h ( [ l .. . . . Em)-' and a = C C Y ~cU=~ Cy,u, . with h ( q l . . . . , nIrn) # 0 then

.

Auf is a. rational function on V defined where f is defined and a --i Aa is linear. If f E P ( V ) and t is an indeterminate then f ( c a t ) can be written in one and only one way as a polynomial in t with coefficients in F. If f is given b y f ( E l , . . . . E r n ) E FIE] then

+

where o ( t ) is a polynomial in t divisible by t 2 . Hence we can determine by the formula

f ( c + at) = f ( c )

+ (A:f)t +o(t).

A: f

(5.4.4)

The fact that A" is a derivation in Q ( V ) implies that Aal = 0 and

5.5. Basic Properties of Reduced Norms

A;f-'

= - ( A ; f ) f ( ~ ) -(if ~ f(c)

# 0).

A consequence of the commutativity of the derivations f in F ( [ ) is

205

(5.4.5)

-i

g.1 5 i 5 n .

n:naf = n:abf .

(5.4.6)

If f E Q ( V ) and f ( c ) # 0 , one defines the logarithmic deri1ratl11e of f at, c and in the direction a by

The11 me have

a: log

n:nnlog

f g = n: log f

f = [f

+ A; log ,9

(5.4.8)

(c)n:na f (n:,f)(n:f)] f (q2. -

(5.4.9)

If II' is a second vector space over F we define a rational map of V into If' as a map p : a -+ C: f,(a)u>,where the ,f, E Q ( V )and the w,E W. These constitute a vector space Q(V,W) over F. ThTedefine

Then

+

+

~:LI, A:(Y 4) = A:(ky) = kA:p. k E F

(5.4.11)

G(fc.1= f ( ~ ) A :+P ( G f ) ~ ( c f) . Q. A irlap y E Q ( V ,1.V) is called hornmgeneous of degree r E Z if for every c at which y is defined and every X # 0 in F, y is defined at Xc and p(Xc) = Xrcp(c). For p E Q ( V ) homo gene it,^ of degree r holds if and or~lyif there exists a homogelleous polynomial g(E1,. . . , <), of degree s + r and a homogeneous polynomial h ( & , . . . :),[ # 0 of degree s such that p is C a , u , * g ( a l , . . . , a n ) h ( a l . .. . , a , ) - ' . differer~tzalequation

For such a function me have Euler's

= T(P(c).

(5.4.12)

In the applications we shall be interested in rational maps into a finite diniensional algebra A. If p and 7) art: two such maps then we can use the product in A t o define y$ by ( p v ) ( a )= y ( a ) Q ( a ) .Then we have the following formula = (A;~)+(C). (5.4.13)

v(c)n:+ +

5.5. Basic Properties of Reduced Norms

+

+

If D is a derivation in an algebra A then D(aba) = (Da)ba ab(Da) a(Db)a. Hence D(U,b) = UD,,,b+Ua(Db). Also D l 2 = D l + 1 ( D l ) + ( D 1 ) 1 = D l + D l = 0. Accordingly, it is natural to define a derzvatzon in a special Jordan algebra H c A to be a linear transformation D of H such that

206

V. Simple Algebras with Involl~tion

W e have seen t h a t i f a , b, c derivation o f A t h e m a p

E

H t h e n [abc] .e

--i

-

[[ablc]E H . Since rc

--i

[abx]

[ax] is a (5.5.2)

is a derivation i n H for any a , b E H . If H is a special Jordan algebra and f E Q ( H ) ;t h e n f is said t o b e Lie inliariant relative to derivations i f AFaf = 0 for every a at which f is defined and every derivation D i n H (Jacobson [68],Section VI.2. p. 220). W e can now state our first m a i n theorem o n reduced norms and rnirlinlunl polynomials o f special Jordan algebras. Theorem 5.5.3. Let H be a special .Jordan algebra, m,,,y(X),t ~ ( a n) ,~ ( a ) the reduced m i n i m u m polynomial, reduced trace and reduced norm respectively of a E H , m the degree of H . Then

(i) t H is a linear function on H and n 3 ~ fis homogeneous of degree m. (zi) n ~ ( a b= ) n H ( a ) n H ( b )if a and b are contain,ed i n a subalgebra F[c]of H . (iii) n H ( X 1 - a ) = m , , H ( X ) . Here n H ( X 1 - a,) is the Jordan, n o r m of X l - a i n H~ (A). (iv) a is invertible i n F [ a ] if and only if n ~ l ( a#) 0 . ('u) Every irreducible factor of r n , , ! ~ ( X )i n F [ X ] is a factor of p,(X) and every root of m , , H ( X ) i n the algebraic closure F of F is a root of p,(X). (vi)

G,H(~= )

(y ) so, i n partics,lar t I i ( l )

=m

and n f I ( l )= 1

jvii) a is nilpotent if and only if ma,H( A ) = x ~ . (viii) If F is infinite an.d D is a derivation i n H then A ~ " T =~0 . ~ (ix) For infinite F we have & T ~ + ~ , H = ( m - i ) ~ , , ~ .

( E l , . . . , En) is homogeneous o f degree 1 i n t h e 5''s Proof. ( i ) is clear since and T ~ , ~ ( <. . ~: En) , . is homogeneous o f degree m i n t h e ['s. ( i i ) Let E l , . . . ,En, 71, . . . , q m , C l , . . . , Cm b e indeterminates. ( u l ,. . . u,) a base for H / F and consider t h e elements x = E,u,, y = C r 17jz"1, z = C y < j x " l , and yz i n H F ( 1 , 7 , , ~W ) . e claim t h a t X = F ( < , q , < ) [ x ]= F ( E , 7 , C)[YI = F(E. v. O [ z ]= F ( t . 11, C ) [ Y ~ ]W. e have

zT

!

m-l

where uJ,(E,7 ) E F [ [ ,71. T h e n d e t ( v J i ) # 0 since t h e specialization 7 2 --i 1, vj -+ 0,j # 2 gives y = x and d e t ( u j i ) --i 1. Hence we can solve (5.5.4) for t h e xi as linear combinatiolis o f t h e yJ so F ( < ,7 ,C)[y]= X . Similarly, F ([! 7 , C ) [ z ]= X . W e now write

-

-

5.5. Basic Properties of Reduced Norms

207

--i 0. k > 1 specializes yz t o x . Since q2 --i 1, qj 0, j # 2 , 1, det(X,,) f 0 so F(E, q , < ) [ y z = ] X. Let u be any element o f X such that X = F(E. rl, c ) [ u ] .T h e n the minim u m polynomial p,(X) o f u in X coincides with the characteristic polynonlial det(X1-L,) where for any a E X , ! ,: v --i av, v E X , since degdet(X1-L,) = m. deg p,(X) = m = deg m , , ~ ( A ) . Hence rn,,H(X) = det(X1 - C,) and n H ( ! u )= det L,. In particular, n ~ ( x =) det L,, n ~ ( y=) det l,, n ~ ( z=) det L, and n H ( y z ) = det C,,. Since X is an associative algebra a C , is a hornornorphism. Hence tV,= lye, and det !,, = (det P,)(det t,). Thus ~ L H ( Y= ~~ ) H ( Y ) ~ H ( z ) . Now let c = CF Y ~ U , , E F and let a , b E F [ c ] .T h e n a = ~y=!' pJcJ,

-

<]

b = C3=0 o J c j .pj, oj E F . KTe have an F-homomorphism o f F[[, 7 . onto F such that ti -, 7%; qJ --i p,, w crj. This extends t o a homomorphism o f HF!c,rl,cl = F[<,q,C]@ F H onto H that is the identity on H . T h e n x c: y --i a , z --i O. yz --i ab under the homomorphism o f H F I E , DIt, ~is~readily . seen also that n H ( z ) n f ~ ( c n) f, f ( y ) --, n ~ ( a n) H : (z) n ~ f ( b a'nd ) n H ( y z ) -i n I I ( a b ) .Hence n H (ah) = n H (a)nH(b)follows from n ~ ( y z=) n ~ ( y ) n ~ ( z ) . . . . ,En, X be indeterminates, x = C t i u i and consider the sub(iii) Let algebra X = F ( [ , X)[x]o f H F ( c , ~ Evidently ). X = F(<,A)[X1- x ] . Hence i f !,, : 1; a u , a , u E X , then as in the proof o f (ii).nH(A1 - z) = det i x l - , = d e t ( i x l - I,) = det(X1 - L,) = m,,H(X). Now let a = Cy a,u, and apply the homomorphism o f H F i E . xinto i H F ( ~such ) that t, --i cri, --, A. Uiider this hoinomorphism z --i a , A1-x --i A1 - a , n ~ ( X -1x ) --i n ~ ( A 1 - a )VL,,H(X) , -i m,.fI(X). Hence n H ( X 1- a ) = m,.Jf(X) follows from nH(X1 - x ) = m,,H(X). ( i v ) \&re define the adjoint of a in H as m- 1

-

C

-

-

-

T h e n m , , H ( a ) = 0 implies that in the subalgebra F [ a ] o f A (the ambient associative algebra o f H ) we have a a # ~= n H ( a )1 = a # ~ a .

(5.5.7)

Hence. i f n H ( a )# 0 then a is invertible in F [ u ]with

W e note next that the multiplicative property o f (ii) implies that n H ( 1 ) 2 = n H ( l )so either n H ( l ) = 1 or n H ( 1 ) = O. In the latter case n H ( a ) = n H ( a l ) = n H ( a ) n H( 1 ) = 0 for all a. This carries over t o AF(X) and implies that m,,H(X) = n H ( A 1- a ) = 0 contradicting the fact that m, H ( A ) is rnorlic o f degree rn. Thus n ~ ( 1=) 1. Now suppose we have a b E F [ a ]such that ab = 1. T h e n n ~ ( a ) n ~ (=b1) and n ~ ( a#) 0. ( v ) I t is easily seen that the first assertion follows from the second. Hence it suffices t o prove the second and by passing t o HF we may assume the base . field is algebraically closed. Now a is invertible in F [ a ]% X r n , , ~ ( X )Thus X I m,,H(X) H X / P,(X). Now for any p E F,p,-pl(X) = p,(A p ) and

+

\'. Sirrlple Algebras with Involutio~~

208

+

+

m,-,~,H(A) = m , , ~(A p) so m a - , l , ~(A) = r n , , ~(A p). It follows that (A p) I m,,H (A) ++ (A p) pa (A). This proves (v). (vi) Evidently ,ul(A) = A - 1. Hence, by (v), ~ z ~ , ~=( (A A -) 1)'". This implies (vi). (vii) If a is nilpotent then ,u,(X) = At. Then m,,H(A) = Am by (v). Conversely, ' v ~ , , ~ ( A =) ATn =+ a is nilpotent. (viii) Let pk denote t,he k-th power map a a k ,k 0,in H and consider the directional derivative Akpk. U7e have Akpo = 0, Akpl = b and we have the recursion formula

+

+

yi

>

+

which follo~vsfrom (5.4.3) by considering (a tb)kt2 = Ua+tb(a + tblk = (U, tUa.6 t2Ub) (a tb) k . From (5.5.9) we can conclude by illduction that

+

+

+

For, A,Dapo = 0 = D l , AFapl = Da and assuming (5.3.10) we obtain from (5.5.9) that

The relation na,,H(a) = 0 can be written in operator form as

Applying A t n we obtain

On the other hand. if .we apply D to the relation mu (a) = 0 we obtain

Subtracting this form (5.5.12) gives

If a is in the open set R of elements a such that pa(X) = ma,H(A) then. by (5.5.14), we have = 0. Then these equations hold for all a E H. (ix) By (5.5.3), A;T%+~,I~is the coefficient o f t in r , + l , ~ ( a + t = ) (-1)"'~ coefficient of tAm-'-l in m , + t l , ~ ( X ) .Since m,+tl.~~(X) = ~ , . H ( A- t ) (as in the proof of (v)) '

= (A - t)"

-

TI,H(O)(X- t)"-l

this is ( m - i ) ~ , , ~ ( a ) .

+ .. . + (-l)"rm,~(a) I?

5.6. Low Dimensional Involutorial Division Algebras. Positive Results

209

As a consequence of Theorem 5.5.3 (viii) we have Corollary 5.5.15. tH([abc])= O for any a. b. c E H . Pmof. Let E be an infinite extension of F and consider H E . Lct a . b E H , c E HE. Then D : c --i [abc] is a derivation in H E . Hence, by 5.5.3 (viii) t H E([abc]) = 0 for a. b E H. c E HE.Since tH, is the linear extension of t~ to H E it follows that we have tH([abc])= 0 for all a , 11, c E H . Another in~portarltconsequence of 5.5.3 is Corollary 5.5.16. For H of degree m we have (i) n I f ( a # ~= ) nH(ajrn-'. (ii) a#"#" (= ( a # i ~ ) # = ~ )rlH (a)m-2a.

The second of these is called the adjoint identity. Proof. It suffices to prove these on the Zariski open subset defined by nH(a) # 0. To prove (ij we apply 5.5.3 (ii) to (5.5.7). Since nH is homogeneous of degree m this gives nH (a)n,H(a#H) = r ~ H ( a )This ~ . gives (i) if nH (a) # 0.To obtain. (ii) me substitute a#" for a in (5.5.7). This gives a#fla#H#H = nH ( a # ~ ) = 1 nfI(a)""'l = nH(a)m-2a#Ha.If n,H(a) # 0, a and a# are invertible in F[a]. Hence we can cancel a # in ~ the foregoing relat,ion to obtain (ii).

SVc can apply Theorem 5.5.3 to the special Jordan algebra H = A' for A any finite dimensional associative algebra. Then mH(X),t H a.nd nH are respectively the reduced lninimurn polynomial, reduced trace and reduced norm in A. Thus 5.5.3 and its corollaries give properties of the reduced rnirlirnum polynomial trace, and norrrl in any associative algebra A. We remark also that since x --i [ax] is a derivation in A, 5.5.15 car1 be improved to t([ab]) = 0 for any a. b E A. Hence t(ab) = t(ba) (5.5.17) llolds for the reduced trace of any associati~ealgebra. We shall see also in Sectioii 5.8 that in the associative case we have the rnultiplicative property n(ab) = rl(a)rl (b) for any a. b.

5.6. Low Dimensional Involutorial Division Algebras. Positive Results Any crntral division algebra D of degree 2 is cyclic (or a quaterrliorl algebra. p.65) so it has the form (E.0. y) where E is a separable quadratic field over F, n the automorpllism # lE of EIF and 7 f 0 in F. The field E = F ( v ) ,u2 = 71 p. D E F. 1 4P # 0. Then a c = 1 - zl and A is generated by v and an element u with the defining relations

+

+

210

V. Simple Algebras with Involution

(,3 = 01, y = 71 # 0 ) . We denote A as ( a ,p]. If char F # 2 we have E = F[w] where u12 = Y # 0 and OW = - w . Then we have the relations

and we denote D as ( a ,7). For any characteristic we have the standard involution Jo : a --i 2 = t ( a ) l - a (5.6.3) where t is the reduced trace. If char F # 2, H ( D . Jo) = F1. Hence the standard irlvolution is of symplrctic type and it is the only symplectic type involution in D. She consider next involutions of second kind in quaternion division algebras.

Theorem 5.6.4 (Albert [39], p. 161). Let D be a quaternion division algebra with center a separable quadratic extension K / F . Suppose D / F has a n in?iolution J of second kind. Then, D = K g F Do where Do is a quaternion algebra over F stabilized by J . Proof. By Theorem 5.3.18.1, D / K contains a quadratic subfield E / K = K % p Eo where Eo is a separable quadratic subfield of H ( D . J ) . Let o be the automorphism # l E , of E o / F . Then a has a unique extension to an a~itomorphismo of E / K and hence there exists a u l # 0 in D such that wvw-' = o v , v E Eo. Applying J we obtain ( J w ) - l v ( J ~=) ov = wvw-l. l i e now take u = u1 + J w if Juj # -w and u = w if JW = -w. Then J u = =tu and uv = ( a v ) u , 2: E Eo. Hence u2 E H ( A ,J ) and u 2 commutes with u and v, so u2 E F* = H ( A , J ) n K * . It follows that Do = Eo + uEo is a quaternion algebra over F stabilized by J and D = K @ F Do. The following converse of 5.6.4 is clear: If D = K % F Do where Do is a quaternion division algebra over F then D / F has an involution of second kind. We obtain next the structure of involutorial central division algebras of degree 4. For this we shall require a part of

Theorem 5.6.5 (Rowen [78]).Let D be a n involutorial central division algebra, K a non-maximal subfield of D , a a n automorphism of K such that a2 = 1. T h e n there exists a symplectic involutzon J and if char F # 2 also a n orthogonal involution J such that J I K = a . Proof. We write J a = a* and J a = a* which is an isomorphism of K onto u * K . This can be extended to an inner automorphism i, of D. Thus we have

5.6. Low Dimensional Involutorial Division Algebras. Positive Results

This implies ( ~ - ~ u * ) - ~ a u - l= u *u*-I (uau-l)u* = ? ~ * - l a ' a u * = ( u ( m ~ ) u - ~=) *( a * a a ) *= (a2a)**= a. Hence u-'u*a = au-'u'

211

and

We can replace u by any non-zero v E U D ~Let . E = *1. Then me claim that there exists a 2: E u D K such that c*-EV # 0. Otherwise. for every d' E D K we have ( u d l ) *= dl*u* = cud'. Taking d' = 1 we obtain u* = EU and dl*u = ud' so d'" = ud'u-I for all d' E D ~ This . implies that D~ is commutative since u - ~(did;)* = dFd;* = ud;d;u-l. Since K for d;, dk E D K we have ' l ~ d ' ~ d / , = is not a maximal subfield of D . D K is not commutative. This contradiction shows that there exists a v E U D such ~ that V * - EV # 0. Then we shall have J a a = a * a = (u* - ~ v ) a ( v-* EV)-' for all a E K and hence

The map a --r (v* - ~ v ) - ' a * ( u *- E U ) is an illvolution of symplectic type if 1 and orthogonal type if char F # 2 and E = -1. By (5.6.8),the restriction of this involution to K coincides with a .

E =

We can now prove

Theorem 5.6.9 (Albert [323],Racine [ 7 4 ] ) .A central division algebra D of degree 4 has an involution if and only if D = Dl @ F D2 where Di is a quaternzon algebra. Proof. The sufficiency of the condition is clear. Now suppose D has an involution. Then D has an involution J of symplectic type (Theorem 5.1.19). By Theorem 5.3.18.3, H ( D . J ) contains a separable quadratic subfield K I F . Let a be the automorphism f l K of K I F . By the foregoing proof there exists a w € H ( D , J ) such that a a = waw-' for all a € K (see (5.6.8)).Since w E H ( D , J ) and w @ F , F ( w ) is quadratic over F. It follows that Dl = K [ w ] is a quaternion algebra over F. Then if D2 = D ~D = ~ Dl, @ F 0 2 . Remark: The quaternion algebra D l constructed in the proof is stabilized by J. Hence D2 = D D 1 is also stabilized by J. Thus the argument shows that if J is a syrnplectic involution then D = Dl @ F D2 where D, is a quaternion algebra stabilized by J . This result for char F # 2 is due to Rowen ( [ 7 8 ] )We . consider next central division algebras of degree 8 with involutions. For these we have Theorem 5.6.10 (Rowen [78]). Let ( D ,J ) be a central division algebra with involution of degree 8. Then D contains a subfield isomorphic to a tensor product of three separable quadratic fields. Equivalently, D is a crossed product ( E , G , k ) where G " Z 2 x Z2 x 22. We remark that at the outset we may assume F infinite and J of symplectic type. For such an involution we put H' = H 1 ( D .J ) = (I J ) D . The proof

+

V. Simple Algebras with Involution

212

we shall give is similar to the proof we gave in Section 2.9 (pp. 69) of Albert's crossed product theorem for degree 4 central division algebras. Again we shall use the Zariski topology to prove the existence of certain elements in H' by proving the existence of corresponding elements in Hb,F the algebraic closure of F. We consider the various instances of this first. We recall that H b is the set of matrices (5.3.16) with r = 4' dt3 t hf2( F ) , d,j = (trd,,)l-d,,'d,, =6,12.6, E F . I f X = ( x i j ) : x , , E M 2 ( F ) ,1 < i , j 5 2 then we write

Any matrix of the form

E H b . These constitute a subspace X of H b . Also let X- denote the subspace of 1kJ8(F) of matrices of the form

and let

Then u E H i and we have

Lemma 5.6.15. If a # @ then the linear maps x --i [ x u ] ,--t~ [ x p u ] ,are bijections of X onto X- and of X- onto X respectively. Proof. Direct verification shows that if x E X then [xu] = [ x - , u] = ( p - a ) x . The result follows.

(a

-

a ) x - and

For x as in (5.6.12) we have

x2 =

(

XX*

0 X1x)

x3 =

(

X*X 0 X * X X0* X )

(5.6.16)

and if x = ( x i j ) xi, , E M2 ( F ) t>hen

XX* =

+

4 x 1 1 )+ 4 x 1 2 ) ~ 1 1 % ~ ~ zalxll x22Zlz n(x21) + n(x22)

+

(5.6.17)

+

We can choose the x,, so that x11F21 x12322# 0. This implies

Lemma 5.6.18. There exist x E X such that x2 is not a diagonal matrix. We prove next

5.6. Low Dimensional Involutorial Division Algebras. Positive Results

213

Lemma 5.6.19. There exists an inuertihle x E X s,(~chthat the nlinimr~m polynomial of x 2 has the f o r m X 2 - aX+P, CY # 0. For such a n x the minim,um polynonzial of x is X 4

-

aX2 + 13.

+

Proof. For x as in (5.6.12) let X = C: &eiz.Then X * = &ell +EleZ2+t4eS3 X X * = X * X = <1<2(e11 e x ? )+&&(e33 e44). Hence x2 is a root of

+

63644:

+

where a = <1<2+<3<4. P = <1[2[3c4.We can choose the ['s so that a # 0. O # 0 and [1<2 # E3&. Then x 2 . hence x . is invertible and x 2 is not a scalar. Hence f ( A ) is the minimum polynomial of x L . Also x is a root of f ( A 2 ) = X 4 -aX2 + D and if this is not the minimum polynomial of x then the form of x , x 2 .x3 implies that r3 = [ x . [ E F . Since x is invertible this gives the contradiction that x2 is a scalar.

-

We recall that if u, v . w E H b then [uvw] 0 (Corollary 5.5.15. p. 207).

[ [ I I u ]E~H ]b

and tnr ( [ ~ L u w=] )

Lemma 5.6.21. There exists x E X . u , .u E Hi, such that ( [ u v x 2[]7 ~ 7 1 5 ] # )@~ F1 where # denotes the adioint in &f&3(~). Proof. B y Lemma 5.6.15 any

has the form [uv]for u , v E H b . For such a y - we have

[uvx]= [y- , x] = [uz!x2]= [ y - x 2 ]=

(Zd i2j

(c*I:

where

Since

where

z,#here is the adjoint in n / f 4 ( ~we ) , have

(5.6.25)

V. Simple Algebras with Involution

214

Now let

Then we have the following formulas:

where

Then

z,#= XlX2X3e14 + XlX2X4e23 + XlX3X4e32 + X2X3X4e41 2.f = P I P ~ P +~ ~~l l I~ ~a m e 2+3~ 1 ~ 3 ~ 4 + e3 ~2 ~3~4e41.

Also

where

9= ~ i ( E 3 E 4- E 1 E 2 ) , u2 = v 2 ( $ 3 $ 4 ~3

= ~ 3 ( $ 1 E 2- $ 3 E 4 ) , u4 = v 4 ( E s E 2

- E1E2): -

E3E4).

Then ( [ u v x 2[] u ~ z ] #is) a~ diagonal matrix whose first two diagonal entries are respectively

It is readily seen that the <'s and 77's can be chosen so that these are distinct. )~ Then ( [ u v . ~ ~ ] [ u v x@] #F1. We can now give the Proof of Theorem 5.6.10. We show first that there exist x , u , v , w E H1 such

that 1. a1 = [ x w w ] is of degree 4. 2. ( [ u ~ a : ] [ u v a l ] #$)F~ l .

These conditions define open subsets of SO it suffices to show that the are non-vacuous. Taking w = u as in corresponding open subsets of (5.6.14) we see that x --i [ x u ~ wis ] a bijective linear transformation of the space X defined above. By Lemma 5.6.19, there exist x of degree 4 in X. Hence

~2")

5.6. Low Diniensional Involutorial Division Algebras. Positive Results

215

there exist a1 = [ x w w ]of degree 4 i11 HI. Also by Lemma 5.6.15 and 5.6.21, there exist .c E X, u , V. U I E H b such that ([~v[zwu~]~][uv[zuiw]]#)~ @ F1. We can llow conclude that we have x , u, v ,w E H' such that 1. and 2. hold. Now suppose x. u. u. w E H' satisfy these conditions. Then, by Lemma 2.9.22. [ ~ ~ v#a0~and. ] as on p. .

By Rowen's lemma (p. 71). F ( a 2 ) is a separable quadratic subfield of D . Let

K denote the involution z,J, z = [ u v a l ] .Then a E H ( D , K ) since by (5.6.31) n = a1 K a l E H 1 ( D ,K ) . \lie now replace the involution J by K and let a E H 1 ( D .K ) be as indicated so a E H 1 ( D .K ) and F ( a 2 ) is a separable quadratic subfield. , ) such that [ ~ a ~$ a ~ ] ~ We show next that there exists an x E H' = H 1 ( D K Fl. It suffices to show that there exists an x E H b such that [xa2a2I4@ F1. We may assume that the ~natrixa2 E Hi, has the form

+

where a1 # a 2 More generally, suppose V is a vector space over a field F equipped with a non-degenerate alternate bilinear form g. Let A = EndFV and let J be the involution in A given by the adjoint relative to g. ( g ( l x ,y ) = g ( x , ( J o y ) .z . y E V). Let H 1 ( A .J) = {P+ J t I k E A ) so H 1 ( A ,J ) is a special Jordan algebra. Suppose a E H 1 ( A .J ) has minimum polynomial with distinct z 5 r , in F. Then there exist r orthogonal idempotents e, in roots a,, 1 H 1 ( A .J) such that C;e, = 1" and a = Ca,e,. Then V = @V, where V , = e,V and the V, are orthogonal relative to g. Hence we obtain a symplectic base for V consisting of symplectic bases for the V, and the matrix of a relative to such a base has the form

<

t,) where m, = [V, : F ] . In matrix form the result is that if a E H'(L',(F). has distinct characteristic roots in F then there exist an invertible symplectic matrix s such that sas-I has the form (5.6.34). I11 particular. this applies to a2 E H b and shows that a2 = diag{allrn,. a21,,). The argument used on p. 70 applied to r n , z , ~ , ( X ) shows that ml = m2 = 4. Then. by Lemrna 5.6.15, any elenlent of x has the form [ x a 2 a 2 ]x, E X. It now follows from Lemma 5.6.19 that there is an x E x such that [ ~ a ~$ aP1. ~ Then ] ~ there exists an z E H' such that [xa2a2I4$ F1.Since H' is a special Jordan algebra, b = [xa2a2]E HI. Since deg H' = 4, the degree of b 5 4. As in the proof of Lemrna 2.9.23 it follows that F ( b 2 ) is a quadratic subfield of D. [a2b2]= 0 and F ( a 2 ,b2) = F ( a 2 )@ F ( b 2 ) . Now put L = z b K Since b E H 1 ( D ,K ) , L is an involution of symplectic type. We replace K by L and let H' now denote H f ( D ,L ) . Evidently b E H ( D , L ) . On the other hand. as on p. 211, La2 = ba2b-I = a 1 - a', a # 0, so a2 @ H ( D , L ) . Then H ( D , L) n F ( a 2 .b2) = F ( b 2 ) .If x E H' and c = [za2a2]

216

V. Simple Algebras with Irlvolution

then Lc = [Lx,La2;Lb2] = [x,a 1 - a2,b2] = - [ma2b2]= -c. Hence c2 E H ( D , L ) . We note also that since [u2b2]= 0, [xa2b2]= [xb2a2]. Suppose there exists an x E H' such that [xa2b2]' = [xb2a2I4 @ F1. Then, as in the proof of Lemma 2.9.23, if c = [xa2b2]then F ( c 2 ) is sepa,rable quadrat,ic over F and [a2c2]= 0 = [b2c2].Moreover, F ( u 2 .c2) = F ( n 2 )8F ( c 2 ) and F(b2:c2) = F(b2) @ F(c2). Also F(c2) fl F ( a 2 ,b2) = F since F ( c 2 ) c H ( D . L) and H ( D ; L) n F ( a 2 ,b2) = F(b2).Hence F(c2) fl F ( a 2 ,b2) c F ( c 2 )n F(b2) = F. Then F ( a 2 ,b2, c2) = F ( a 2 )% F ( b 2 ) @ F(c2). Thus the proof of Theorern 5.6.10 will be complet,e if we can show that there exists an I(: E H' such that [xb2a2I4@ F1. By Zariski topology this will follow if we can show there is an x E H i such that [xb2a2]' @ p1. AS before, me may assume

where

Dl f /3% Then the condition

[a2b2]= 0 implies that

where a, E 12.f4(F).We claim that we rrlav assurne that

where a1 and a 2 are the roots in F of the minimum polynomial p(X) of a2. a 1 - a 2 .a # 0.a: = a14 - a t where the * is defined by (5.6.11). Since La" Now let V be a four dimelisional vector space over F equipped with a nondegenerate alternate bilinear form g. Let l, be the linear transformation in V with matrix a, relative to a given syrnplectic base for V and let l',denote the adjoint of i, relative to g. Then = a l v - h,. We have the decomposition V = Vl @ Vz where V i is the eigenspace of k , relative to the eigenvalue a k . Suppose u and u E V,,

!:

Hence, if g(u, v) # 0, then a, = a - a, and 2ak = a. Then char F # 2 = 1/20 E F contrary to the irreducibility of p(X) in FIX]. Thus and g ( u . 71) = 0 and Vk is totally isotropic. Then [Vk : F] 5 2 : and since V = Vl [/19 V2, [Vk : F] = 2. Let (ul. u2) be a base for V,, IF. Then there exist a base (vl. va) for V, such that (ul, c l , u2, ~ 2 is) a symplectic base for V and the matrix of ti relative to this base is diag{al, a 2 : a l , a2). Hence there exists an invertible symplectic matrix si such that s,a,s;l = diag{al, a 2 , a l , an).Then s = cliag{sl. ss) is a symplectic matrix that commutes with b2 and transforms s into diag{al, az, a l , a 2 , 0 1 , a2,01. 02).

5.6. Low Dimensional Involutorial Division Algebras. Positive Results

217

Now let x be of the form (5.6.12) where

then [xb2a2I2= diag{y2,y 2 , 0 , 0 , 0. 0 , i 2 , Y 2 )where 0. Hence [xb2a2I4@ Fl arid the proof is complete.

=

(PI - P z ) ( a ~-aa) #

Our next result will show that involutorial central division algebras of degree 8 are close to being tensor products of quaternion algebras. This is the following Theorem 5.6.38. If D is an involutorial central division algebra of degree 8 then &12(D) is a tensor product of four quaternion algebras.

This theorem was proved by Tignol [78] for characteristic # 2. The exterisiorl to characteristic 2 is due to Rowen [78].We derive first some lemmas required in the proof. Lemma 5.6.39 (Albert [39], Th. 10.16). Let D be a division algebra with center K a separable quadratic extension of F and suppose D has an involution of second kind and a subfield E / F of D I F such that E K E E @ F K . Then there exists a,n in~iolu~tion J' of second kind i n D .~'uchthat E c H ( D , J ' ) .

Proof. Write Ja = a*. We have the isomorphism J I E of E onto E'. This extends to a n isomorphism of the field E K / K = (E @ F K ) / K into E * K / K . Since E K is a simple subalgebra and D / K is central simple the ison~orphism of E K / K can be realized by an inner automorphism of D . Thus there exists a g # 0 in D s~lchthat a" = g a g p 1 , a E E. (5.6.40) Then

a = (g*) - la i g* = ( g * ) - l g a g p l g * ,n E E

(5.6.41)

so g-lg* E D E = D E K . If g-'g* + 1 = 0.g" = -g and we take J' = i,-I J . On the other hand, if h = g-lg* + 1 # O then g* + g = gh E H ( D , J ) and h E DE. In this case we take ,J' = Z [ ~ + ~ * ) -JI. In both cases J' is an involution of second kind such that E c H ( D . J ' ) . Lemma 5.6.42. Let D be a quaternion algebra satisfying the conditions of Le~nnla5.6.39. Then D = D l @ F K where D l is a quaternion division algebra over F containing E .

Proof. B y Lemma 5.6.39, we may assume E C H ( D , J ) . Then the proof of Theorem 5.6.4 carries over.

218

V. Simple Algebras with Involution

Lemma 5.6.43. Let D be an involutorial central division algebra of degree 4 , E = El E 2 a subfield of D / F such that E , / F is separable quadratic. Then D = D l @ F D 2 where Di is a quaternion algebra containing E,. Proof. By Lemma 5.6.5 the non-trivial automorphism of E 2 can be extended to an involution J of D / F . This stabilizes DE2 and J / DE2 is an involutioil of second kind. Since DE2 > E , and E l E 2 = E l @ p E 2 we can apply Lemma 5.6.42 to obtain DE2 = E l @ F D2 where D2 is a quaternion algebra containing 17 E 2 . Then D l = D ~ >ZE l and D = Dl @ F D 2 . Let A = ( E ,a, a ) where E is separable quadratic, a is the non-trivial automorphism in E and 0 # a E F . Then we can choose a generator v of E such that T E I F ( v )= 1. Then v 2 = v + p, 8, E F , and mv = 1 - v. Hence A is generated by u and a second element u such that

We denote.A as ( a ,p] . Then we have

Lemma 5.6.45. If A = ( a l ,D l ] is a division algebra and ( a l ,B l ] 2 ( a 2 ,Pz] then there exists a ,01E F such that

, we may assume we have generators ui,v i , i = Proof. Since ( a l ,Dl] ( a z pz] 1 , 2 , for A such that u: = ai, v t = V , +P,, uivi = ( 1 - vi)u,. Suppose first that F ( u l ) # F ( u 2 ) and ulu2 u2u1 # 0. Then

+

+

so ulw,2 U Z U ~centralizes u l and similarly it centralizes u2. Since F ( u 1 ) # F ( u 2 ) ,F [ T L u2] I : = A and hence ulu2 uaul = y E F and 7 # 0. Now

+

Hence if v' = r - 1 u 1 ~ 2then

We have

u2v1= r-1u2uluz = y - l ( y = ( 1 - v1)u2

-

uluz)u2

u l v l = U I ( ~ - ~ U I U=~u)l ( y - l ( y - ~ = u 1 ( 1 - y - l u 2 u 1 ) = u1 - vlul = (1- vl)ul.

2 ~ 1 ) )

Hence A = ( a 1,Or] , = ( a 2PI] , and (5.6.46) holds in this case.

5.6. Low Dimensional Involutorial Division Algebra?. Positive Results

219

Our proof will be completed by showing that there exist a # 0 such that F ( u l ) # F ( a u 2 a - I ) and ulau2a-' a ~ ~ a - ~# u 0.l Let 0 = { a I a # 0 and F ( u 1 ) # F ( a u 2 a p 1 ) ) ,O r = { a I a # 0 a,nd ulau2ap1 +au2ap1ul # 0). We claim that 0 and 0' are open subsets of D. This is clear of 0' and it follows for 0 by observing that the defining condition is equivalent to [ul!uu2ap1# ] 0. Hence to prove that t,here exist a such that F ( u l ) # F ( a u a a - l ) and u l a u 2 a p 1+ a u 2 a p l u l # 0, it suffices to show that 0 # 8 and 0' # 8. To see that 0 # 0 we may assume F ( I L ~=) F ( u 2 ) . Then let u i = v2u2v,I. Then F ( u 1 ) # F ( u h ) . Ot,herwise, F ( u 1 ) = F ( u 2 ) = F ( u h ) and i,;, / F ( u 2 ) is an automorphism. Since ug = az,i,",uz = &uz. On the other U ~ . u2v,l-u2 = &u2 hand, i,,u2 = u2u2v,l = ~ ~ ( l - - v ~=) Uv ~y ~~J Y ~ -Hence so either ' u ~ =u 0 ~or 2u2 ~ = u 2 v a 1 .Both of these are impossible. Hence either F ( u I )# F ( u 2 ) or F ( u 1 ) # F ( u & )and O # 8. To see that 0' # 0 we may assume F ( u l ) # F ( u z ) since we can replace u2 by uf2 = 7 ~ ~ 2 ~ 2 1to 1 ; achieve ~ this. We may also assume char F # 2 since if char F = 2 then ulu2 u2u1 = 0 implies uluz = u2u1 SO F ( u 1 ) = F ( u 2 ) contrary to hypothesis. Hence u1u2 u2u1 # 0 and 1 E 0'. It remains to uau1 = 0 . show that 0' # 8 if char F # 2 , F ( u 1 ) # F ( u 2 ) and ulus Assume these conditions. Then it suffices to show that if F is the algebraic closure of F then there exists an invertible a in M 2 ( F ) = DF such that ulau2a-I au2a-IuI # 0. To see this, by replacing ul by a similar matrix, we may assume

+

+

+

+

+

Then ,ulu2 = - 7 ~ 2 ~ 1implies , that u2 has the form

Using these determinations of u l and u2 it is readily seen that there exists an invertible a in M2 ( F ) such that ulau2a-l au2a-'ul # 0. Hence there exists an a # 0 in D such that F ( u l ) # F ( a u z a - l ) and u l a u 2 a p 1 au2ap1ul # 0. Then we may replace u2 and v2 by au2ap1 and av2ap1and obtain the situation we considered first.

+

+

If char F # 2 it is preferable to use another generation of quaternion algebras than the one we have used in 5.6.35. To obtain this from the generation Then we obtain the defining by u , 11 as above we replace u by u, = v relations u 2 = a , w 2 = -(.uw = -wu (5.6.47)

i.

i.

where y = P+ We now use the not ation A = ( a ,7)to display the parameters in this generation. Then we can formulate Lemma 5.6.35 for char F # 2 as follows.

Corollary 5.6.48. I f A = ( a l ,7 1 ) is a division algebra and (al,nIl) ( a 27, 2 ) then there exists a y' E F such that

V. Simple Algebras with Involution

220

We can now give the Proof of Theorem 5.6.38.. Let D be an involutorial central division algebra of degree 8. By Rowen's theorem 5.6.10, D contains a subfield E = E l C$F E 2 @JFE3 where E, is separable quadratic over F. Put Ey = K and consider D' = D K . Then D' D K so D' has exponent 2 in B r ( K ) . By Albert's theorem D' has an involution as algebra over K . Also D' contains E{ = Ell( and E 2' - E 2 K and E{Eh = E i @ K Ea. Hence, by Lemma 5.6.43, D' = D i @ K DL where Di is a quaternion algebra over K containing the subfield E,'. Then D: = (Ei, o,, a , ) / K where a, is the automorphism of E,'/K whose restriction to E, is the nontrivial automorphism of E,/F. By Theorern 5.6.5. the non-trivial automorphisnl of K / F can be extended to an involution J in D / F . Then J stabilizes D' and J D' is an involution of second kind in D r . By Theorem 5.2.7. the existence of such an involution in D' implies that cOrKIFD1 1 in B r ( F ) . Hence

-

-

and by Theorem 3.13.21

Hence ( E l : 01: N K / F ( ~ ~@F ) ) (E2; ~

2 N, K

(5.6.50)

/ F ( ~ ~") 1 )

where we have written a, for C T ~ Ei. Using a generator v, for E, such that u,2 = u, Pi we can write (Ei, a,, 1VKIF(ai)) = (ATKIF(ai),Pi]. Thus (NK/F( ~ 1 )PI] ) @ F (NK/F(a2), ,021 1 and corlseyuently (A'K/F ( a l ) ,PI] (NKIF(a2),Pa]. By Lemma 5.6.45, there exists a P' E F such that for n, = NKIF(ai) we have

+

"

-

--

Then (n,. 8,] @ (n,,P'] 1 and ( n l , p'] @ (n2. P'] 1. Suppose first that char F = 2. By Theorem 4.8.13 we obtain (n,. P, + P'] 1, 1 = 1 , 2 and by the usual formula for tensor products of cyclic algebras, (nln2,O1] 1. It now % K follows from Lemma 5.2.10 that the algebras (a,, P, +PI] and ( ~ 1 ~P']2 over have involutions of second kind. By Theorem 5.6.4, each of these algebras is a tensor product of K with a quaternion algebra over F. On the other hand, ( a ~,% ,

+ P'l @ K (a,, ,Pa + P'l @K (ala2,

-

-

PI]

( a i , Pi1 8 ( a l , P'l @ (az, Pal % (al. ,Dl] 8 (aa, P'l (a1,Pll BK (a21 Pal = D; @,Di = D l .

5.7. Some Counterexamples

221

Thus A f 2 ( D 1 )and hence M 2 ( D )contains a tensor product of three quaternion algebras over F. Then A f 2 ( D )is a tensor product of these and the centralizer which is also a quaternion algebra. In the case char F # 2 we use the standard generation of quaternion algebras. Then we can write ( n , , P,] = (n,,7 , ) and we have a 7' such that

-

which implies that (n,.7 , ~ ' ) 1 and (n1n2. 7') is the same as that in the char 2 case.

1. The rest of the argument

5.7. Some Counterexamples In this section we shall give a construction due to Amitsur, Rowen and Tignol [79]of a central division algebra of degree 8 with involution that is not a tensor product of quaternion algebras. Suppose E I F is finite dimensional Galois with Galois group G = (a1)x . . x (a,) where a: = 1 SO [ E : F ] = /GI = 2T. Let U = ( u i j / 1 I i , j I r ) ,u t j E E" , satisfy U,, = I , 1L:l = 'U. . . 23 37 (5.7.1) ( a k ' U . i j ) ( a i ' U . j k ) ( ~ j u k i= ) UzjUjkUkz

(5.7.2)

N i ( N j ( u , , ) ) = 1, N i ( a ) = a ( a i a ) .

(5.7.3)

As in Section 4.6. we construct a non-commutative polynomial ring E [ t ;a . U] whose elements can be written uniquely in the form C a,,...

i,t y

. ..t p .

i . . .E

OIij
(5.7.4)

and -\?rehave the relations

<

E [ t ;a , U ] is a dornairi with center C = F[<], = ( E l , . . . , [,), ti = a,tP where a , is an element of E*, determined up to a multiplier in F*, such k 5 r (Theorem 4.6.33). The central lothat a i ( a k a i ) - l = N i ( u k i ) ,1 calization E [ t ;a : UIc* is an abelian crossed product ( E ( [ ) a, , U.a - 1 [ ) : a = (n1> . . . . o r ) aP1E , = ( a l l < l ,. . . , over its center F ( < ) ,the field of fractions of C . We recall also that E [ t ;a , U ] was constructed by iterating the t,wisted polynomial construction beginning with D(') = E [ t l ;all and defining D("') = ~ [ t ( " ' ) ; u ( ~ + as' ) ]the twisted polynomial extension ~ ( ~ ) [ t k&+I] + l ; where Ck+i is the automorphisln in D ( ~such ) that

<

222

V. Simple Algebras with Involution

Kow let

T

E G ,E = (

,

E ~ . ,. . E ~ ) E. , = i

l . Then we have

Theorem 5.7.9. There exists an involution J,,, of E [ t ;a , U ] such that if ancl only if T U , ~= a i a g u J z .1

(5.7.11)


Proof. Suppose J,,, exists. Then applying this to (5.7.7) gives ~ , ~ ~= t ~ t , ( E , ~ ~ , ) ( E ~ ~ , ) ( T U Hence ~,). t 3 t z = t , t j ( ~ u i 3and; ) by (5.7.7), t,ti = uLLj,titg= t , t , ( a , a J u J , ) .Hence we have (5.7.11). To prove the converse we require

Lemma 5.7.12. Let R be a ring with involution J : a --, a*, a an autornorphism i n R. Then J can be extended to an involution J In the twisted polyrlomlal ring R [ t ;a ] so that t* = i t zf and only if a ( a a ) * = a*, a E R.

(5.7.13)

Pmof. If the extension exists then for any a, b E R and k.! E N uTe have ( a t k ) * = ( f t ) " * , (bte)* = ( & t ) q h "and ( ( a t k ) ( b t e ) ) "= ( a ( a k b ) t k f e ) ' = ( i t ) k + e ( a k b ) * a *On . the other hand. ( ( a t k ) ( b t e ) ) * = (bte)*;(atk)* = ( f t)'b* ( i t k ) a * = ( ( f t ) k + e ( a p k b * ) a Hence *. (akb)* = a-"*. In particular. we have (5.7.13). Conversely, assume (5.7.13). Then. by induction, we obtain ( a k a ) *= opkaa"for all I;. Retracing the steps of the foregoing argument we see that this implies that if we define then we have ( ( a t k ) ( b t e ) ) = " (bte)'(atk)*and J is an anti-automorphism in D [ t .n] extending J on D and mapping t --, i t . Then J is an involution. To complete the proof of Proposition 5.7.9 we note that for any T E G we have 01 ( r a l a ) = r a . Hence we have an involution in E [ t l .o l ] whose restriction to E is T and t l --i ~ ~Nowt suppose ~ . k < r and we have an involution Jr,E(k) in D ( ~such ) that J7,,(k) I E = T and J,,,(k)t, = ~ , t , , 1 z k. Let hk+l be the autonlorphisrn in D ( ~satisfying ) (5.7.8). We claim that (5.7.13) holds for ,J,,,~L, and a = ak+1. Since E and the t,, 1 i k, generate D ( ~it) suffices toverify (5.7.13) f o r ~ = E ~ + ~ E , aa E n d a = t,, 1 s t k . Thefirst isclear. For the second we have for E , = *1.

< <

< <

<

5.7. Some Counterexamples

223

Hence (5.7.13)holds for t, and 5.7.9 follows by induction. The involution J,,, we have constructed has a unique extension to E[E;r. UIe* = ( E ( [ )a, , U, a-1[) which is a central division algebra over F(El,...,ET). In the remainder of this section we assume char F # 2. In this case we shall derive a condition that a central simple algebra A is a tensor product of quaternion algebras. For this we require Definition 5.7.15. A subset S = {u,) of A is a q-generating set if ( 1 ) u,2 E F*. ( 2 ) For any u,, 7 ~ ,E S either I L , ~ , = I L , U ~ or uiuj = -uju,i. ( 3 ) If u, # u, then there exists a uk E S that commutes with one of the pair {ui,u,) and anti-commutes with the other. (Note that this is satisfied trivially if u,uJ = -uju,.)

We remark that (3) is redundant if S is a base for the central simple algebra A. Proposition 5.7.16. (i) Any q-generating set S is linearly independent. (ii) If (u1,up) (ul # up) is a q-generating set then so is { l , u l .u 2 ,ulup) and Q = F Fu1 Fup Fulup is a quaternion algebra.

+

+

+

Proof. (i) If S = {u,) is not linearly independent then we have a relation C: a,u, = 0, a, E F* and k minimal for such relations. Evidently I; > 1. By (3) of the definition there is a u E S such that either [uul]= 0 and [uup]# 0 or [uul]# 0, [up]= 0. It suffices to consider the first case. Then k 0 = [u,Cazuz]= C,=2 a, [uu,].Since [uu?]= 0 or [uu,] = 2uu,, we have a relation 2 ~ ( za:,u,) (,> summed ~ on some of the indices in ( 2 . . . . , k ) including 2. Since char F # 2 and u is invertible this gives a relation C1aJu,= 0 shorter than the original one contrary to the minimality of 5 . (ii) By (3).we have ulup = -uzul. The two assertions are an immediate consequence of this. We can now establish the following criterion. Theorem 5.7.17. Let A be central simple of degree 2'. Then A is a tensor product of quaternion algebras if and only if A contains a q-generating set S of cardinality 4,.

+

+

+

Proof. Let A = Q1 @ F . . . @F Q, where Qi = F1 Ful, Pupi F u l , ~ p i where uii = pli # 0 and uliu2i = -upiulz Put u ~ = i I , u3i = uliuzi Then 5' = { ~ i , l u z , a ...uir, 1 0 5 ij 5 3) is a q-generating set of 4, elements of A. Conversely, suppose A contains a q-generating set S of 4' elements. Then S contains U I , u2 such that u1uz = --upul. Then Q1 = F+ Ful +Pup Fu1u2

+

224

V. Simple Algebras with Involution

is a quaternion subalgebra of A. Hence A = Q 1 ( A Q 1 ) Q1 8~AQ1 and the centralizer AQ1is central simple of degree 2'-'. The factorizatiorl A = QIAQ1 ilrlplies that every element of A has the form a = a0 alul a2ua asulua where a, E AQ1 and since A 2 Q1 % AQ1 this representation is unique. Let u E S and suppose u @ AQ1.Then we can not have uul = ulu and 1~uz= u ? ~ . Hence we have the following possibilities:

+

+

+

i 71U1 = - I L ~ U , uu2 = uyu ii ~ L U I= - I I , ~ ~ L , 'uu2 = -uzu iii 11u1= ulu, uu2 = -uyu

+

+

+

and u = a0 alul azu? ajulua,a, E A&'. In the first case one sees that a0 = a1 = as = 0 so u E AQ1uz.In ii we obtain u E AQ1u3 and in iii (L E AQ1ul.Thus S = u : = ~ ( sn AQ1ut)if uo = 1 and us = u1u2. No-' AQ1uZ is a subspace and [AQ1u,: F] = [AQ1: F ] = dr-l. Since the elements of S are linearly independent it follows that 1 S n AQlu,I = 4 ' ~ ' .In particular, the set AQ1n S has cardinality 4'-l and hence is a base for AQl. Then this is a q-generating set in the central simple algebra AQ1by the remark following Definition 5.7.15. Then. by induction on the degree. AQ1 Q2 8 . .. 8 Q,. Q, quaternion, so A 2 Q1 @ . . . 8 &,.

"

Again consider the abelian crossed product ( E ( [ )o, , U,a-l[) which is the central localization of the iterated twisted polynomial ring E [ t . a. U]. For such an algebra the criterion of 5.7.17 yields the following Theorem 5.7.18. If ( E ( [ )u, , U ,aC1<)is a tensor procluct of quaternion algebras (over F ( [ ) )then the algebra contain,^ a q-generating set of the form

Proof. 1%observe that ally element of a q-generating set can be replaced by a rnultiplc of it by a non-zero element of the center. Hence if ( E ( [ )o, , U,a p ' [ ) is a tensor product of quaternion algebras then it has a q-generating set S of hence of elements of the form cardinality 4' consisting of elements of E [ t ,a, U]: .f = Eni ,...i,.ty . . . t:, a,,... iT 6 E,i, > 0. For such an element we define the leading term X ( f ) = ai ,... irtz,' . . . tZ,. with ai ,... i". # 0 and ( i l , .. . , i,) maximal in the lexicographic ordering of N ( ' ) . Then it is clear that if f # 0. g # 0 then . f ~f 0 and X ( f g ) = ( X f ) ( X g ) . Hence i f f g = * , 9 f then ( A f ) ( X g ) = f( X g ) ( X f Also if f E C = F[<]then X ( f ) E C. It follows that if S is a q-generating set, then X(S). the set of leading terms of the elements of S , is also a qgenerating set. Hence we may suppose that the elements of S have the forrn ai t y . . . t;:. Now i j = 2qj + i(, where ii = 0 or 1. Since t; = a y l t j , multiplication of ail ..., ?t: . . . t: by a central element permits us to assume that every element of S 11a.sthe forrn ail ...,rt'y. . . t: , 0 5 i j 1. Since IS = 4' it follows that S is as in (5.7.19).

1.

<

W-e now specialize to the case in which r = 3. Then E = El @ E2 @ Eg where E, = F ( q , ) ,q: = P, E F" and n, is the automorphism such that o,qi =

-q,> o,q2 = q 7 , j # i . We take T = 010203. TITe suppose we have a nlatrix U = (u,,, 1 5 i , g 5 3). utl, E E* such that (5.7.1)-(5.7.3)and (5.7.11) hold. We can simplify these conditions by putting

Since u,'

= 2 ~ j , ; for

any

T

E S3 we have

Now we claim that = l >1 5

Nz(tlj)

For (5.7.22) \ire note that

T

i 5 3%

= f l T 1 0 K 2 0 T 3 , 7i E S3 and by (5.7.11)

Hence ATn3 ( ~ ~=3 1)which is (5.7.22). Also (5.7.22) implies c,v, = v,'.

by (5.7.2)

1

v1

(5.7.22)

-1 ,-1 113 -

u2

Then.

(01~1)(02~2)(03%'3)

Hence ( V ~ U = ~ 1Vand ~ )(5.7.23) ~ holds. Conversely, assume we have vl, va; v3 E E* such that (5.7.22) and (5.7.23) hold. Define ui, = 1 and define u,j for i # j by ~ ~ 1=. v ~z y 2. This is well defined since there is a unique .ir E S3such t,hat (?rl,7r2)= (i: j ) . Clearly u i l = uji. Thcn reversing the order in (5.7.24) TVC obt,ain (5.7.11). Also (5.7.22) and (5.7.23) yield (5.7.2) for distinct i, j.k. Since (5.7.2) follows frorn (5.7.1) if any two of i , j,k are equal, we have (5.7.2) for all i. j,k. Now we have

Hence (5.7.3) holds. Thus all the conditions (5.7.1)-(5.7.3)and (5.7.11) on the u , ~are consequences of conditions (5.7.22) and (5.7.23) if we define u,, = 1 and uTl,2 = Now we have defined a, above (p. 220) to be an element of E* such that a,(oka,)-l = N,(uk,). 1 k 3. Hence if we put b, = a,'. then

UZ.

< <

We shall now sllow that

226

V. Simple Algebras

with Invol11tio11

First, we have bi(aibi)-' = Nl(u,,) = 1 so aibi = b, and if the v's are defined as before, then

Hence ~ , ~ b= ,aT3bTl ~ and aT2aT3bTl= b V l . Thus bTl E Inv a,l n Inv a,~a,3 = F(q.rr2q.rr3). Since N n s ( v T 3 )= 1 there exists y E E' such that v,3 = and bince aT2bTl= aXSbTl

This implies that u l = b,lN,l(ySg") E Inv 0,s. Also w E Inv u,l since bTl and N,l(ySg") E Inv a,l. Thus w E Inv a,l n Inv a,3 = F ( q T 2 ) Hence . 2 1 , ~ = ? L ~ N , ~ ( ~E ~F(q,l)N,l(E). ~ " ) - ~ If we take T = 1 and T = (23) in this relation we obtain bl E F ( q z ) N l ( E )n F(q3)N1(E).This and our previous result that bl E F(q2qs) implies (5.7.26). Next we shall turn things around and prove Proposition 5.7.27. Suppose .cue have a non-zero element b in F(q2qs) n F ( q 2 ) N 1( E )n F(q3)N1( E ) . Then there exists vi, 1 i 3, satisfying (5.7.22) and (5.7.23) (and hence (ui, 1 5 i , j 5 3) satisfying (5.7.1)-(5.7.3) and (5.7.11)) such that if ( b l , b 2 , b s ) are elements of E' satisfying (5.7.25), then we can take bl = b.

< <

Proof. We can write b = w2Nl(y2) = w ~ N ~ (where ~ B w, ~ E) F(q,) and y, E E. Let v2 = (a2y;l)y3, us = (03y,l)~2,vl = ( ~ ~ ~ 3 Then ) - l . (5.7.23) holds and N2 (v2) = 1 = N 3 ( v 3 )Also

since w2 E F ( q z ) and hence a3w2 = w2. Since b E F(q2qa),azb = gab. Hence

Similarly, N 1 ( v 2 )= (02b)b-' = (a3b)b-I. Then N l ( u l ) = N ~ ( U , ~ U ~ ' ) = 1. Hence we have (5.7.22) and we can define ( E ( < )a, , U, for U = ( u , ~ ) , u,i = 1,urrl,,2 = u z y and the bi such that (5.7.25) holds. Since b E F(q2q3), b y hypothesis, and bl E F(q2q3) by (5.7.26), bbrl E F(q2q3) and hence al(bb,l) = bbll. Also

so a2(bb,l) = bbyl. Similarly, 0 3 ( b b , l ) = bbll. Hence b b l l E F' and we may replace bl by b in ( b l , b2,b 3 ) .

5.7. Some Colinterexamples

227

\Ve also have t h e following criterion for ( E ( [ )a, , U,a - l [ ) w i t h r = 3 t o b e a tensor product o f quaternion algebras. Proposition 5.7.28. Let A = ( E ( 6 ) .a , U , a - 1 [ ) be the central localization of the iterated twisted pokyn,omial rin,g E [ t ,a . U]where r = 3 and assume A is a tensor product of quaternion algebras. T h e n bl = a,' E F N 1 ( E ) . Conversely ass11,me that A / F ( [ ) h,as an involution, and bl E Fi"\il(E). T h e n A is a tensor product of quaternion algebras.

Proof. I f A is such a tensor product, t h e n , b y (5.7.18) A has a q-generating set t h a t includes a n element a t l , a E E*. T h e n ( ~ tE F~( E)) * .~O n t h e other hand, ( a t 1 ) 2 = N l ( a ) t : = N l ( a ) b l & . Hence N l ( a ) b l E F ( [ ) n E = F and bl E F N l ( E ) . Conversely, suppose A is involutorial and bl E F N 1 ( E ) . T h e n bl N l ( a ) E F* for sorne a E E*. T h i s implies that ( q l ,a t l ) is a q-generating set. T h e n Q 1 = F ( [ ) F ( 6 ) q l F ( [ ) a t l F ( [ ) q l a t l is a quaternion subalgebra o f A. T h e n A = Q 1 @ AQl is central simple o f degree 4. Also, since [AI2= 1 i n B r ( F ( E ) ) ,[AQ1I2 = 1 . T h e n AQ1 is involutorial and AQl is a tensor product o f quaternion algebras b y Albert's theorem (5.6.5). T h e n A is a tensor product o f quaternion algebras.

+

+

+

Sire have now reached one o f our main goals: a field theoretic criterion for t h e existence o f a central division algebra w i t h involution o f degree 8 that is not a trnsor product o f quaternion algebras. Theorem 5.7.29. Let E be a n abelzan extenszon of degree 8 of F wzth Galozs group G = ( 0 1 ) x ( ~ 2 X) ( 0 2 ) , S O E = F ( q 1 ) 8 F(q2) @ F(q3) where a,q, = -q,. a J q q,~ ~f J # 2. Suppose there e x ~ s t sa non-zero element bl E E such that

T h e n there ezists arL involutorial central division algebra A of degree 8 over F ( & , 6 2 %E3): ti indeterminates, that is not a ten,sor product of quaternion algebras. Proof. Given bl satisfying ( i ) , b y Proposition 5.7.27, we have tl,. 1 5 i 5 3, such t h a t (5.7.22) and (5.7.23) hold and hence a matrix U = ( u Z 3 / 1 5 i,j 5 3) satisfying t h e conditions for defining t h e abelian crossed product, A = ( E ( ( ) ,a . U , up'<) and a n involution J i n A. W e know also t h a t A is a division algebra o f degree 8 over its center F(E1,( 2 ; E3). B y Proposition 5.7.28, i f bl satisfies (ii) t h e n A is not a tensor product o f quaternion algebras.

If S is a subset o f a finite dimensional extension field K I F we shall denote t h e { N I C I F ( s )I s E S ) b y N K I F ( S ) .Note that i f L is a finite dinlensional extension field o f K and S is a subset o f L t h e n N L I K ( N K I( S~ ) ) = N L I F( S ) .

228

V. Simple Algebras with Involution

Using these notations we can state two useful variants of the conditions (i) and (ii) of Theorem 5.7.29 as follows. Proposition 5.7.30. If b E F(qzq3) then b E F ( q z ) N l ( E )fl F(q3)ATl(E)zf and only zf

N2(b)= N 3 ( b ) E hT~(y~,y~)/~(q,)(F(ql.q2))

(5.7.31)

&F(ql. q ? ) / ~ ( q ~ ) ( ~q(3q) )l ..

Proof. If b E F(q2qy)then a2b = ash and N2(b) = N3(b). Suppost b E F'(q2)i\Tl(E).Then b = cn;l(d).cE F(q2),d E E , and N3(b) = N3(c)AT3(hi1(d)) = c2Nl(Ai3(d))= Nl ( c N s ( d ) ) Since . Gal E / F ( q l , q 2 )= ( g 3 ) ; AT3(d) E F(q1,q2) and cA!3(d) E F(q1. q 2 ) Since Gal E ( q l ,qz)/F(qa)= ( 0 1 1 F ( q l ;q2)).N I ( ~ A ~ '=~N(~ ~ ( q )l ~)y 2 ) / ~ ( q ~ ) ( c A r 2 ( d ) ) .

Sinlilarly, if b E F((13)Nl( E ) then

Con~'ersel~. let lv3(b) ~ V ~ ( q l r q 2 / ~ (F(q1, ( q 2 ) q2)), SO F(q1,qa). Put f = d b , g = d ald E F(q2). Then

+

+

1?\T3(b) =

Nl ( d ) .d E

# 0, this gives b = ( b + g + a 3 b ) - ' N l ( f ) and b+g+asb E F ( q z )sinceg E F(q2) and b ash E F(q2) since b E F(q2,q3) and Gal F(q2. q3)/F(qZ)= (c3/ F(q2. q3)).If ,f = 0 then b = -d E F(q2,q 3 ) n F ( q lq2) , = F ( q a ) .Tlms in either case b E F(q2)1yT1 (E). Similarly, if N2(b) = N3(b) E A T ~ ( q l , q 3 )(/F~((qql .qg)) J) then b E F(q3)N1( E ) .

Iff

+

Proposition 5.7.32. If b E F(q2q3)n FAT1(E)then

Proof. We prove first that

Both brackets on the right hand side are contained in F(qzq3)and in N 1 ( E ) since Gal F(qzq3. ql)/F(qzq3) = ( 0 1 / F(q2q3% q l ) ) and Gal F(qaq.3,qlqa) /F(q2q3) = ( ~ / 1 F(q2q3,qlqz). Hence the left hand side contains the right hand side of 3.7.34. Now let a = N l ( c ) E F(q2q3).If c E F(q2q3,ql) then

5.7. Some Counterexamples

229

Nl(c) E N~(q~q~.~~)/~( q lq) )~since ~~) Gal ( ~F(q2q31 (q2q qi)/F(qzq3) 3~ = (PI I F(q2q3, q l ) ) . Hence in this case a = N l ( c ) is contained in the right hand side of (5.7.33). Now suppose c @ F(q2q3,q l ) . Since (1,qa) is a base for E / F ( q 2 q s , q l ) ,c = ci(c2 + qa),ci E F(q2qs;q i ) . Then

+ q2) = N I ( e l )(c2 + q2)(ale2 + qa) = N i ( c i ) ( N l ( c a )+ qp(c2 + + 9;) E F(q2q3) since N l (c,) and q; E F(q2q3),qz(c2 + 0 1 ~ 2 )E F(q2q3). Then q2(c2 + a l e s ) E Inv(a2as). Since m a 3 (92(c2 + = -q2 (c2 + e2 + 0 1 ~ = 2 0. a = N l ( c ) = iV1 ( e l ) %

( ~ 2

0 1 ~ 2 )

0 1 ~ 2 ) )

0 1 ~ 2 ) .

Since c2 E F(qzq3. q l ) which is quadratic over F(q2q3)with base ( 1 ,q l ) and Gal F(q2q3. ql)lF(q2q3) = (a1 I F(q2q3, q l ) ) it follows that cz = w q l , w E F (q29s). \&re now have

This proves (5.7.34). Now let b E F N l ( E ) n F ( q 2 q 3 ) .Then b = a u , a E F , a E lCT1(E)n F(q2q3). B y (5.7.34)

Hence 1 v F ( q 2 q 3 ) / ~ ( b )= c ~ ~ N F ( ~ z ~ ~ ) / F ( ~ )

E

[-hJ~(q,q3)/~(N~(q2y33y~)/~(q2q3)(F(q2q3,ql))l

(5.7.35)

Now the first bracket in (5.7.35) is 1CTF(q2q3,ql)/~(F(q2q3)ql) = NF(~~),F ( N F (q 1~) /~~ (~q l )~( F ( q 291) q 3C. N F ( ~ ~ ) / F ( FSimilarly. ( ~ I ) . the second bracket C h r ~ ( q l(q F (aq )l q/2~) ) .But also, the second bracket C N F ( ~ ~(F(q2qs). ~ ~ ) ~ F Hence (5.7.32) is a consequence of (5.7.34). We are now ready to give the Amitsur-Rowen-Tignol example of a n insrolutorial central division algebra of degree 8 that is not a tensor product of quaternion algebras. As base field we take F = Q(X), X a n indeterminate, and q2 = q3 = It is we take E = F(q1. qz. q3) where ql = readily seen that E / F = F(q1) 3 F(q2) 3 F(q3) and hence E is a n abelian extension of F with Gal E I F = ( a l ) x ( 0 2 ) x ( a s ) where a,q, = -q,, a,q3 = qJ if J # 2 . m7e shall show that

n. d m ,

but

bl

!2 FN1 ( E l .

a.

230

V. Simple Algebras with Involution

Then it will follow from Theorem 5.7.29 that there exists an involutorial central division algebra of degree 8 over F ( ( 1 , E2. E 3 ) = Q (A. E l , ( 2 , t s ) , A. tzindeterminates, that is not a tensor product of quaternion algebras. To prove (i) we note that. by Proposition 5.7.30. it suffices to show that lV2(~2q3)= -q;qz = X ( X 2 1) E

+

This holds since

To prove (ii) it suffices to show, by Proposition 5.7.32, that

On the contrary, suppose

where f l E F ( q 1 ) . f 2 E F ( q l q 2 ) . f 3 E F(q2q3). We can obt,ain relations in Z [ X ] from these equations by writing

where g and the g, E Z [ X ] . Then we have

whicli gives the equations

-

Hcnce it suffices to show that these equations for the g, E Z [ X ] are impossible. If we have g, E Z [ X ] satisfying (5.7.36) we may suppose the g.c.d. of the g, is 1 . We now apply the homomorphism of Z [ X ] onto Z / ( 2 ) [ X ]such that X A, a --i u = a + ( 2 ) for a E Z . Thenwr obtain X(X+1)2(ij1+g2)2 = ( ~ ~ + ( X + l ) g=~ ) ~ g ~ + X ( X + 1 ) 2 g ~If. gl+g2 # 0 then the degree of the first term is odd while that of the middle term is even. Hence gl = 9 2 , ,g3 ( A l ) g 4 = 0 = g j = j g , again by degrees. Hence 2 1 g5 and 2 1 gg. Then 4 1 (g;+X(X+l)g,"). Then, by (5.7.36). g;) and 4 1 (9: - (A2 1)g:). Since = ga, 92 = g1 2h1, hl E Z[X] 4 1 (g: and g; g; = 2912 4glh1 4 h f . Since this is divisible by 4, gl is divisible

+ +

+

+

+

+

+

+

5.7. Some Counterexamples

231

by 2 and g2 is divisible by 2. Also since g3 = (A + l ) g 4 , g3 = (A + l ) g 4 + 2h3 and 4 [(A 1 ) 2 g i 4(X l)h3g4 4 h i - (A2 l ) g i ] SO 4 1 2 942 and 2 / g4. Then also 2 / g3. Thus every g, is divisible by 2 contrary to the choice of the g,. This contradiction establishes (ii) and completes the proof of

+

+

+

+

+

Theorem 5.7.37). There exists a n involutorial central dzuision algebra of degree 8 over Q(X, E2, E3), X and the Ei indeterminates that is not a tensor product of quaternion algebras. We recall that if D is a quaternion division algebra with center a separable quadratic extension K of the base field F and D has an involution J of second kind then D = K @$F Do where Do is a quaternion subalgebra stabilized by J (Theorem 5.6.4). On the other hand. the Amitsur-Rowen-Tignol example can be used to prove

Theorem 5.7.38. There exists a division algebra D of degree 4 over its center K , a separable quadratic extension of the base field F , with involution J of n stabilized by second kind such that ( I ) D contains n o q ~ ~ a t e r n i osubalgebra J , (2) D is not a tensor product of K and another subalgebra stabilized by J . Prooj. Let A be the involutorial central division algebra of degree 8 constructed in the proof of Theorem 5.7.37 and let K = F ( q l ) ,D = A K . We have an involution J of A stabilizing K and mapping ql into -91. Then J stabilizes D and J / D is an involution of second kind. If D contains a quaternion subalgebra Q stabilized by J then A = Q 8 AQ and AQ is involutorial central simple of degree 4. Hence by Theorem 5.6.4, AQ is a tensor product of two quaternion algebras and A is a tensor product of quaternion algebras contrary to Theorem 5.7.37. This proves ( 1 ) . Now suppose D = K @ B where B is a subalgebra stabilized by J. Then B is central with involution of degree 4 so B is a tensor product of quaternion algebras and AB is quaternion. Then A is a tensor product of quaternion algebras. Amitsur. Rowen and Tignol have also given in [79]an example of a central division algebra of degree 4 with involution J that contains no quaternion subalgebra stabilized by J. They have shown that such an algebra exists over the field Q(X, E l , E 2 ) , X and the 5, indeterminates. We shall not give their example. Instead we shall give in the next section a criterion due to Knus, Parimala and Sridharan that a central simple algebra of degree 4 with involution contains a quaternion subalgebra stabilized by the involution. This criterion has been used by Saltman to construct examples of central simple algebras of degree 4 with involution J containing no quaternion subalgebra stabilized by J over any Hilbertian field (Saltman [ 9 2 ] ) .

232

V. Simple Algebras with Involution

5.8. Decomposition of Simple Algebras with Involution of Degree 4. Definition 5.8.1. An algebra with involution (A. J ) is said to be ciiecomposed if A = A1 @ F A2 where the A, are stabilized by ,J and [A, : F] > 1. In this section we shall consider the decomposition of central simple algebras of degree 4. We have seen that if ( D , J ) is a central division algebra of degree 4 with a symplectic involution then D decomposes as a tensor product of two quaternion algebras stabilized by J (see the Remark on p. 210). \Ire shall now extend this result to central simple algebras.

Theorem 5.8.2. Let (A, J ) be a central simple algebra of degree 4 with a symplectic in,volution .J. T h e n (A. J ) decomposes. Proof. Since we have proved this for A = D a division algebra it remains to prove it in the two cases (i) A = M 4 ( F ) and (ii) A = ,bf2(D) where D is a quaternion division algebra. (i) The classical result that any tu70 alternate matrices in ,W2,(F) are cogredient (e.8. BA I , p. 352) implies that any two (A[z,(F), J ) with Jsymplectic are isomorphic. The result now follows by taking ( M 4 ( F ) .J ) = (11f2(F) 8 1W2(F),t @ t s ) . (ii) Here we consider Endo(V) where V is a two-dimensional vector space over the quaternion division algebra D rather than 1Vf2(D).It can be seen from the results of 5.1 that symplectic involutions in A = EndD(V) are given as the adjoint map relative to a non-degenerate hermitian form g on V with respect to the standard involution a --i ti in D . It is known that for such a form there exists a base ( u l , u2) of V over D such that the matrix of g relative

to this base has the form

(7

:2)

, a , t F' (Jacobson. l o c cit; p. 133).

Then (EndoV, J ) is isomorphic to (hf2 (D), K) where

Now 1Vf2(D) is a tensor product of the subalgebras D = {diag{d. d ) / d t D ) and n'12(F) and each of these is stabilized by K. Hence (A/r,(D). K) and (Endo V.J ) are decomposed. We shall give next a criterion due to Knus, Parimala and Sridharan [91] that a central simple (A, J ) of degree 4 decomposes. For such an algebra we define the subspace S(A, J) = (1 - E J ) A (5.8.4) where E = 1 or -1 according as J is of orthogonal or symplectic type. By extending the base field to its algebraic closure one sees that [S(A, J ) : F] = 6.

5.8. Decomposition of Simple Algebras with Invollition of Degree 4.

233

For the sake of simplicity we assume F infinite in the remainder of this section.

Proposition 5.8.5. There exists a linear map p~ o,f S ( A ,J ) lnto itself such that U P J ( ~= ) p ~ ( a )E a F l , P? = ~ ~ s ( A . J6) E , F* (5.8.6)

for a E S ( A ,J ) . Moreover, p~ is unique up to a non-zero 7nultzplier zn F .

+

Proof. Suppose first that J is of symplectic type. Then S ( A ,J ) = ( 1 J ) A is a special Jordan algebra. Let n and # denote the reduced norm and corresponding adjoint in S ( A ,J ) (p. 206). Since the degree of S ( A ,J ) is 2 , a# is a linear function of a and n ( a ) is homogeneous quadratic in a. We have aa# = n(a)1= a#a. By 5.5.16(ii),we have a## = a. To prove the uniqueness assert,ion we observe that if p~ is any polynomial map satisfying (5.8.6)then a is invertible in S ( A ,J ) if and only if p , ~ ( a )# a 0.Passing to the a,lgebra,icclosure p of F we can apply the Hilbert Nullstellensatz to conclude that p J ( z ) z and n ( z ) have the sasne irreducible factors (exercise 3, p. 432 of BAII). Since n ( z ) is irreducible and p J ( z ) is linear it follows that p J ( x ) x = pn(x) where p E F*.Then p J ( x ) x = px#x. This implies that pJ(a)= pa# on the Zariski for all a. open set of invertible elernents. Hence pJ(a)= Next let K be of ~rt~hogonal type. Then S ( A ,K ) = ( I - K ) A . Let J be any involution of symplectic type. Then K = i,J where Ju = -u which is equivalent to K u = -u. If a E S ( A ,,J) then uu and a 7 ~ ~E' S ( A ,K ) . Hence we have the bijective linear maps a --, ua and a --, aupl of S ( A ,J ) onto S ( A ,K ) with inverses b --, u-'b and b -i bu. Now define for b E S ( A ,K ) p ~ < ( b= ) (u-lb)#upl E S ( A ,K ) .

(5.8.7)

Then pK(b)is linear in b since a# is linear in a. We have

= b p ~ ( b ) Next . we prove that since (u-'b)(uP'b)# E F1. Similarly, p ~ ( b ) b E F*. Suppose first that u-'b is invertible in S ( A ,J ) or. equivalently, n,(uP1b)# 0. Then (u-'b)# = n(u-lb)(u-lb)-' so p ~ ( b = ) n(u-'b)(u-'b)-'u-' = n(u-lb)bP'. Then p&(b) = n ( ~ ~ - l h ) ~ K ( b -=l ) n,(wplb)n(uplbpl)b. Now bu = c~(u-'b)u= -(Ju)(u-lb)u. Hence n(bu) = n A ( u ) n ( u p ' b )follo~vsfrom the formula for the reduced norm in S ( A ,J ) ) and the well-known fornllsla for Pfaffia.ns of alternate matrices: (see p. P f ( ( t U ) Y U )= (det U ) Pf(Y), Y alternate, U arbitrary. Thus n(uP'b) = n , ( ~ ) ~ ' n ( b uand ) p&(b) = n(u-'b)n(uplb-')b = n ~ ( u ) - ' n ( b u ) n ( u ~ ' b ~ ~ ) b = n ~ ( u ) - ' b Hence . p& = 61s(n,K)with 6 = n ~ ( u ) - # l 0.The restriction that u-'b is irsvertible ca,n be dropped from this result since the set of these elernents is open in the Zariski topology. The uniqueness property of p~ for K of orthogonal type follows from this property for simplectic type by using t'he passage from S ( A ,K) to S ( A ,J ) that we have noted.

p''< = S I S ( A , K ) , 6

234

V. Simple Algebras with Involution

Now let ph = p p ~p , E F*. Then we have p',(p&(b)) = n A(uf)-'6. Since ph is linear and homogeneous and p;( = p p this ~ gives p2nA(u)-1b= nA(ul)-lb.Hence n A ( u l )= p2nA(u).Thus the multiplier nA(u)-' is determined up to a multiplier that is a non-zero square in F. We note also that any ~ = n A (u)-lb. For. given such invertible u can be used in the relation p~ ( p (b)) a u we can define J = 2,-IK which is an involution of symplectic type such that K = 2, J . Our result shows also that for any invertible K-skew elements u and u'. nA(u)-' and n ~ ( u / ) - differ ' by a multiplier in F*2.We shall call the element n A ( u ) ' F * ' of F * / F * 2the dzscrzmznant S ( K ) of the orthogonal involution K . We now have the following Theorem 5.8.8. Let A be central simple of degree 4 with involution K of orthogonal type. The71 ( A ,K ) decomposes if and only if 6 ( K )is trivial (= F*').

Proof. Suppose ( A ,K ) = (A1C3FA2, K l g K 2 ) where A, is a quaternion algebra with invohition K t . Let t l be the standard involution in A1, t2 an involution of orthogonal type in Az. Then t l 8 t 2 is an involution of symplectic type in Al @ A> Let u, be an invertible element in A, such that Kt = z,,*t,. Then K = z,,g,,(tl 8 t 2 ) and hence 6 ( K ) = n ~ ( u ) - ' F *where ~ u = ul 8 u2. Aforeover, n A ( u )= n A( u l ) n A ( u aand ) n ~ ( u ,=) nAt ( u , ) ~Hence . n ~ ( uis) a square and S ( K ) is trivial. For the proof of the converse we first determine 6 ( t ) ,t the transpose involution in A = AJ4(F).Here S ( A .t ) consists of the matrices of the form

and We now put

VCTehave aa' = P f ( a ) l= a'a and a --, a/ is a linear transformation in S ( A ,t ) . It follows that pt is a --, a'. Now suppose K is an orthogonal involution in A such that S ( K ) = 1. Then we may assume that p ~ ( p ~ ( b=) b) for b E S ( A ,K ) = ( 1 - K ) A . We now define S ( A , K ) += {a E S ( A , K ) I p ~ ( a=) a ) (5.8.12) Since upK( a ) E F1, we have a2 E F1 if a 6 S ( A ,K ) + . Now S ( A ,K ) + is a subspace of S ( A ,K ) / F . To determine its dimensionality we extend the base , ) ( M ~ ( F t)),. It field to the algebraic closure I@. Then we obtain ( A F K

5.9. Multiplicative Properties of Reduced Norms

235

-

follows that [ S ( A I<)+ , : F] = [S(11/f4(F), t ) + : F ] . It is clear from (5.8.9)and (5.8.11)and the fact that pt : a a' that S ( M ~ ( Ft)+ ) , is the set of matrices of the form / 0 a12 a13 a14 \

Hence [S(n'f4( F ) ,t ) + : F] = 3 and [ S ( A K , ) + : F] = 3. Now S ( M (~F ) ,t)+has the base ( I L = el4 - e23 e32 - e41;u = e l 3 e24 - e31 - e42>w = el2 eal e34 - ess) and

+

+

+

+

+

Hence if char F # 2 then F1 S ( M * ( F )t,) + is a quater~lioilsubalgebra of Atl,(p)stabilized by t . Since (F1 S ( A ,K ) + ) FN F1 S ( A J ~ ( Ft ))+, ,F I S ( A ,K ) + is a quaternion subalgebra of A. Clearly this is stabilized by K . Now assume that char F = 2. If ( A ,K ) = (il14(F), t ) , then, by (5.8.13), B = S ( A ,K ) + F1 has the base (1,u, w ) where u = e l 4 e23 e32 e41, v = e l 3 e24 e31 e42, w = el2 e21 e34 e43. Then u2 = v2 - w2 = 1, uv = w = z?u,uw = I L = %UV,uw = v = wu SO B is a commutative Frobenius algebra. It follows by a base field extension argument that B = F l + S ( A , K ) + for any ( A ,K ) , K of orthogona,l type, is a con~nlutativeFrohenilis algebra. Since [B : F] is the degree of A, any derivation of B into A can be extended to an inner derivat'ion of A (Theorem 2.2.3, ( 3 ) ) .Let 11 and v be generators of B and let A be the derivation of B into A such that Au = u , Av = 0. Then there exists a d E A such that du ud = u. dv = ud. Then d 2 d E AB = B (Theorem 2.2.3, ( 1 ) ) .Hence (d2+d)' = cul. Put f = d2. Then f 2 + f = a1 and u f + f u = u o r f u = u ( f + l ) .It isreadily seenthat C = F l + F f + F u + F f z ~ is a quaternion subalgebra of A. Now K u = 1~ and ( K f ) u u ( K u ) = u SO ( K f f ) u = u ( K f f ) and ( K f f ) u = u ( K f f ) . Hence K f f E B and ( K f f ) 2 E F 1 T h e n K f 2 + f 2 E F and ~ K f 2 E C . Since f 2 + f = a 1 it follou~sthat K f E C and KC = C. Hence C is a quaternion subalgebra of A stabilized by K .

+

+ + +

+

+

+

t i ,

+

+

+ + +

+

+

+

+

+

+

+

+

+

+

5.9. Multiplicative Properties of Reduced Norms In the remainder of this chapter we shall be concerned with the special Jordan algebras H ( A . J ) of simple associative algebras with involution ( A ,J ) . Lie shall be interested in further properties of the Jordan norm of H ( A . J ) and in relations between H ( A ,J ) and A and ( A ,J ) . In this section we derive the important multiplicative properties of reduced norms for arbitrary special Jordan algebras H (with finite dimensional ambient A ) . Our results on these will be based on a special case of the Hilbert Nullstellerlsatz that if V is an ndimensional vector space over an algebraically closed field F and f ( E l . . . . , ),[

236

V. Simple Algebras with Involution

and g(EL,.. . , En) have the same zeros in V then. as noted in Section 5.8, these polynomials have the same irreducible factors in F [ t l , .. . A second basic tool in our discussion will be the elementary theory of inverses in special Jordan algebras. For arbitrary Jordan algebras (not necessarily finite dimensional) we have the following

.I,[.

Definition 5.9.1. An element a special Jordan algebra H is znvertible if U, is bijective on H . Then the znverse of a is a-I = Ua- l a . The following result gives the elementary facts on invertibilit,~.

Proposition 5.9.2. I. The following conditions are equivalent: (i) a is invertible. fii) There exists a b € H such that U,b = a , U,b2 = 1. Then b = a-'. (iii) 1 E U,H. 2. If a is ,invertible so is a-' and U,-I = U,-I. 3. a and b are invertible H U,b is invertible. In this case (U,b)-I = Ua-IbF1. 4 . If a is invertible so is an, n 1, and (an)-' = ( a - l ) n . O n the other hand, ,if an is invertible for some n > 0 , then a is invertible.

>

Proof. The proofs follows from the identity (5.3.2): LT,UbUu = UUab 1. (i)+(ii). If U, is invertible there is a b such that U,b = a. Then U, = UClab= UaUbUawhich implies that Ub = U l l . Then U,b2 = U,Ubl = 1 and h = U,-'a = a-I. (ii)+(iii). We note first that in any associative algebra A the condition that a is invertible with inverse b is equivalent to the Jordan conditions U,b = a. U,b2 = 1. For if these hold we have aba = a , ab2a = 1. The second implies invertibility of a. Then the first implies that b = a-I. Conversely. if ab = 1 = bu then aba = a and ab2a = abba = 1. Now U,b = a and U,b2 = 1 imply UaUbUa= U a , UuUb2Ua= 1 and since U p = Upbl = UbUIUb= G U , = 1. Hence applying the foregoing remark to A = EndFH we see that U, is invertible with Ub = U;'. (iii)+(i). If (iii) holds we have a b such that Uab = 1. Then U,UbU, = U1 = 1 and U, is invertible. 2. We have shown in the proof of 1. that U,-I = U,-'. Hence U,-I is bijective and a p l is invertible. 3. Since UUab = U,UbUa and U,UbU, is invertible if and only if U, and Ub are invertible, it is clear that U,b is invertible if arid only if a and 11 are invertible. In this case (U,b)-' = ( U U a b-'Uab ) = (UaUbUa)-'Uab = u,-~u~-~u,-~U b = ~ - 1 u - l b= U-lb-1 a

b

a

4. The induction definition of a n , n imply that U,n = U:. This implies 4.

> 0. and the fundamental formula

If H is special and finite dimensional then U, is bijective on H if and only if it is either injective or surjective (which can be replaced by the weaker condition 5.9.2 1. (iii)). If U, is not injective then there is a b # 0 in H such that U,b = aba = 0. In this case we say that a is a zero dzuzsor in H . It follows that if H' is a subalgebra of H and a E H' is invertible in H then a is

5.9. LIultiplicative Properties of Reduced Norms

237

invertible in H 1 and a-I E H i . On the other hand, if a is invertible in HI then the element condition 5.9.2 1. (iii) shows that a is invertible in H. Also we have seen in the proof of 1. that if A is associative then a is invertible in A in the usual serise if and only if a is invertible in A'. These remarks imply that if H is special and finite dimensional. then a is invertible in H if and only if a is invertible in the associative algebra F [ a ] .Hence by Theorem 5 . 5 . 3 4. we have the following important criterion.

Proposition 5.9.3. A n element a of a finite dimensional special Jordan al(lebra H is invertible i n H i n the sense of Definition 5.9.1 if and only if & ( a ) # 0. We can now state the main theorem on the multiplicative properties of reduced norms.

Theorem 5.9.4. Let n~ be th.e reduced n o r m of a special Jordan algebra H of dimension n and degree m. T h e n

if a a,nd b are contained i n a subalgebra B of H that is a subalgebra of the arr~bier~t associative algebra A of H . For the proof we shall require

Lemma 5.9.5. Let V be a n n-dimensional ~uectorspace with base ( v l : . . . , v,) over a n algebraically closed field F . Let g ( & ; . . . <,,) be a non-constant polynornial i n F[E] = F [ [ I , .. . and let p , ( [ ~ :.. . .E,,): . . . . p , ( [ ~ ,. . . , En) be the distinct prime factors of g(E1,. . . , E n ) i n F [ [ ] (determined u p t o m,ultipliers i n F * ) . Sr~pposeT is a bijective linear trarlsformation of V that maps the hypersurface Z ( g ) = { a € V I g ( a ) = 0 ) onto itself. T h e n there exists a permutation . i ; o f { l , . . . .r ) a n d p 3 E F * , l < j < r ; s u c h t h , a t

Proof. Let Tz12 = C r z k v k a, = Ca,z~,.Then T a = Ccu:v, where a: = C r k 2 a k .If f ( E 1 , . . . , E n ) E F[EI we define f T ( E 1 , . . . . E n ) = f ( E : . . . . , [A), Ei = C ~ k ~ E l c . The11 f ( [ I .. . . , 6,) --i f T ( [ ~. ., . En) is an autoinorphisrn of F [ E ] . Hence f ([I, . . . . En) is prime if arid only if f ( [ I .. . . , En) is prime and hence the distinct prime factors of g T ( [ l , . . . , En) are P T ( [ ~ ,. . , En). . . . , . . . , En). Now g T ( a ) = g ( T a ) for a E V and our hypothesis is that g ( a ) = 0 g ( T a ) = 0. Thus g(E1. . . . , En) arid gT ( [ I , . . . , &,) have the same zeros in V. Hence. by the Hilbert Nullstellensatz they have the same prime factors in I?[<]. Hence we have a permutation .rr of (1,.. . , r ) and p, E F* such that

.

PT(E1,

238

V. Simple Algebras with Involution

Then (5.9.6) holds. We recall that n H ( l ) = 1. A polynomial having this property will be called 7~ormalzzed.It is clear that a non-constant normalized polynomial has a unique factorization as a product of normalized prime polynomials. We can now give the Proof of Theorem 5.9.4. 1%':may assume the base field F is algebraically closed. (i) Let pl (El: . . . , En), . . . ,pr(El , . . . , En) be the distinct normalized prime factors of nH(E1,. . . , t n ) in F[E].For a E H let G, be the group of invertible elements of F [ a ] .This is a non-vacuous open subset of F [ a ] (defined by n ~ (c) [ ~#] 0). Hence it is connected in the Zariski topology. For any c E G,, the map U, is a bijective linear nlap of H. By Proposition 5.9.2 (iii), this stabilizes the set of invertible elements of H : which, by Proposition 5.9.3, is ) 0. Hence U, maps the hypersurface defined by the set defined by n ~ ( b # I L ~ ( X=)0 onto itself. Then, by Lemma 5.9.5, there exists a permutation ~ ( c ) of ( 1 , . . . , r ) and a p3 (c) E F* such that

Now consider the map c --, U, of G, into the group GL(H) of bijective linear transformations of H . If e l , c2 E G, and b E H we have

Hence c --t U, is a homomorphism. It follows that c --i ~ ( cis) a hornomorphism of the abelian group G, into S, and ker n is a subgroup of finite index in G,. This is the subgroup of elements c E G, such that

If we put b = 1 and recall that p, (1) = 1 we obtain p, (c) = p, (c2). Hence the conditions that c E ker T are

These are polynomial conditions on the coordinates of c, hence they define a closed subset of G,. Thus ker T is closed in G, and since it is of finite index in G,, ker T is also open. Since G, is connected it follows that ker T = G, and hencp (5.9.11) holds for all invertible c E G, and all b E H. By continuity we have (5.9.11) for all c E F [ a ] and all b E H. It now follows that we have

for all a,b E H. On the other hand, by Theorem 5.5.3 (ii), n,H(a2)= nH(a)'. Hence we have (i).

5.9. rY1ultiplicative Properties of Reduced Norms

239

(ii) Suppose ( ~ 1 ,. .. , u,) is a base for H such that ( v l , .. . , v q ) is a base for be a normalized prime factor of n H ( E l , . . . , Eq, 0 , . . . , 0 ) . and let p(E1; . . . , suffices to prove the multiplicative property for p(El; . . . , & ) . Let n B ( E 1 , .. . , E,) be the reduced norm for B. Tl~erinB( E l , . . . ,[,) and n H (El , . . . , Eq, 0,. . .) have the sarne zeros ill B, hence t,he same prime factors in F [ E 1 , . . :& ] . Hence it suffices t,o assume H = B and prove the multiplicative property for the norma,lized prime factors ~ ( ( 1 ,. .. , E n ) of n B ( E 1 , .. . , t,). The argument for . . . , Cn); . . . : pr(E1, . . . :tr) this is similar to the proof of (i). As before let be the normalized prime factors of n B ((1, . . . ,6,) and for a E H let G, be the group of invertible elements of F [ a ] .For c E G, we now let c~ be the bijective linear map x --i cx of B. Then replacing U, by c~ in the foregoing argument we obtain p, ( c ) E F * such that

c,)

and putting b = 1 we obtain p,(c) = p:,(c) and

Then this holds for all c , b E

B and hence n ~ ( c b=) n ~ ( c ) n ~ for ( b all ) b. c E H .

The multiplicative properties given in this theorem essentially characterize the reduced norm of a special Jordan algebra and of an associative algebra. For, we have

Theorem 5.9.16. (i) Let H be a special Jordan algebra of dimension n and se . . , E n ) is a raormalized homodegree m over an infinite field F . S ~ ~ p p o g(E1,. geneous polynomial # 1 i n FIE] such that

for a , b E H . T h e n g is a product of normalized prime factors of n H ( x ) = n ~ ( E 1 :. . ; E n ) . (zi) Let A be an associative algebra of dimension n and degree m over a n ir~finitefield F ; g(E1,.. . ,En) a normalized homogeneous polynomial # 1 i n F[E] such that d a b ) = g(a)g(b) (5.9.18) for a , b E A. T h e n g is a product of normalized prime factors of nA( E l , . . . ,En). Proof. (i) We shall use the hypothesis (5.9.17) only for any a and b = : g ( U u a # ~= ) g ( a ) 2 g ( a # H )Since . this holds for all a in H over an infinite field we have ~ ( u , z # H= ) g ( x ) 2 g ( x # H for ) x = Cy ELuL.Now U,X#H = X X # H X = nH( x ) x since X X # H = n~ ( x )1 in F [ x ] .Hence

and if deg g = q we have g ( n H ( x ) x )= n ~ ( x ) q g ( x Hence ).

240

V. Sirnple Algebras wit,h Involution

Cancelling g(z) we obtain n,H(z)4 = g ( z ) g ( x # ~ )Evidently . this implies that the normalized prime factors of g(z) divide n ~ ( z ) . (ii) As in (i) we obtain g ( x ) g ( x # ~ = ) g(zx#~= ) g(nu(s)l) = n ~ ( z ) q which implies (ii).

Remark: The arguments apply if IF1 is finite but sufficiently large. For (i) it suffices to have F > (deg g)2 and for (ii). IF/ > deg g suffices. It is readily seen that nlf(E1,. . . ,En) for H = H ( A >J ) for (A. J ) central simple with irlvolution is prime except in the case char F = 2: J of symplectic type. In the exceptional case the norm polynomial nH1 ((1; . . . , En) for H1(A,J ) = (1 J ) A is prime. For the associative central simple A the primewas an immediate consequence of the classical result ness of nA (€1,. . . ,),[ that det(Ei,) for indeternlillatjes ti, is prime. In the same may the result on H ( A , J ) and H1(A,J ) follow from this and the following easily proved results: 1. The determinant det(tZj)where <,3 = [j, is prime in F[El1,. . . ,<, I. 2. The = 0. EZj = -[,, is prime in F[E12,. . . , (,-1,,,] is Pfaffian Pf(<,j) where prime. A consequence of these results and of Theorem 5.9.16 is

+

cii

Corollary 5.9.19. ( i ) Let (A: J ) be central simple with involution over an, infinite F and g(E1,. . . , En) is a normalized homogeneous polynomial # 1 i n n indeterminates where n = [H(A,J ) : F] unless char F = 2 and J is o,f symplectic type i n wlziclz case n = [H1(A,J ) : F].Suppose we haue (5.9.17) for all a , b E H ( A , J ) unless char F = 2 and J is of syrnplectic type i n which case we assume (5.9.17) for all a , b E H1(A,J ) . T h e n g is a power of nH(E1,.. . ,En) i n the non-exceptional case and of nHt (El;. . . ,En) if char F = 2 and J is of symplectic type. (ii) Let A be central simple associative of dimension n over a n injinite F and g(E1, . . . , [,) is a normalized llomoger~eouspolynomial i n n indeterminates such that (5.9.18) holds for a , b E A. T h e n g is a power of nIi(tl, . . . , En).

5.10. Isotopy and Norm Similarity Let (A. ,J) be an associative algebra with involution ( J : a --i a * ) and let c be an invertible element of H(A. J ) . Then c defines the involution J(') = i c J in A and since the condition J(')a = a is equivalent to (c-la)" = c - l a , the map cL1 is a bijective linear map of H(A, ~ ( ' 1 )onto H(A, J ) . We have = c-' and for a. b E H ( A , ~ ( ' 1 ) .

till

5.10. Isotopy and Norm Similarity

241

To explain these equations we introduce the concept of isotopy for special Jordan algebras. We observe first that if c is an invertible element of an associative algebra A then we can introduce a c-multzplzcatzon in A by acb = acb.

(5.10.2)

This 1s associative with unit c-' so it defines tlie associative algebra A(') with c-multiplication as multiplication and c-' as unit. We call A(') the czsotope of A. Since c-lab = (c-'a)c(c-'b) tlie map c ~ isl an isomorphism of A onto A('). Now consider A(')+. This has the unit c-l and U-multiplication a. b --i a,b,a = acbca = U,U,b. We write u:') = U,U,. This is the U-operator in A(')+. Nore generally, let H be a special Jordan algebra with A as ambient associative algebra so H is a subalgebra of A+ arid suppose c and c-I E H . Then H is a subalgebra of that is, the subspace H of A contains the unit c-I of A(')+ and is closed under ~ ( ' 1 : if a. b E H . We deriote this special .Jordan algebra with ambient algebra A(') by H(') and call it the c-rsotope of H . IIoreover, we shall call an isomorphism of a special Jordan algebra H onto tlie cl-isotope of a special .Jordan algebra H' an zsotopy of H onto H'. Thus q is an isotopy of H onto H' if and only if there exists an invertible element c1 of HI such that q1 = el-', v(U,b) = U,,Uc~(7jb) (5.10.3) Using this definition we see that (5.10.1) states that c i l is an isotopy of H(A. J(')) onto H ( A , J ) . Though A and A(') are isomorphic it is not difficult to glve examples in which H(A. and H ( A , J) are not isomorphic. To obtain an example we note that any matrix algebra A = &&,(D), where D has an involutioii 3, the Jordan algebra H ( A , J ) , J the standard involution and H(A. ~ ( ' 1 ) . J(') the canonical involution determined by the diagonal inatrix c = diag{c,. . . . , c,) and 3. where ?, = c, is invertible, are isotopes. This is readily verified. On the other hand examples exist where these Jordan algebras are not isomorphic. For instance, take D = R, the field of real numbers. and el = -l.r, = . . . = c, = 1. The Jordan algebra determined by the standard involution has no nilpotent elements, while that detrmined by the canonical involution does have such elements. There is an alternative definition of isotopy due to Koecher. Let H and H ' be special Jordan algebras. Then a map q of H into H ' is called an zsotopy (zn Koecher's sense) if 7 is bijective linear and there exists a bijective linear map q* : HI --i H such that for all a E H

First. suppose rl is an isomorphism of H onto the cl-isotope of H', so we have (5.10.3). If we put U,l(qb) = b' so b = q - ' ~ ; ~ b ' , then. by (5.10.3), we have U,,bl = q ~ , ~ - ' ~ ; l b 'SO we have (5.10.4) with v* = v - l ~ ~ Conversely, l . suppose 7 is an isotopy in Koecher's sense. Then U,(l) = qq* is bijective of H' onto H' so q(1) is invertible in H'. Let c' = q(l)-' and put a = 1 in (5.10.4). This gives qq* = U,(l) and q* = It follows that we have (5.10.3) and

242

V. Simple Algebras with Involution

so q is an isomorphism of H onto ~ ( " 1 . We state the result we have proved as Proposition 5.10.5. If H and H' are special Jordan algebras then a map q of H into H f is a n isom,orphism of H onto som,e cf-isotope of H f if and only if q is bijective linear and there exists a bijective linear map q" of H f onto H such that (5.10.4) holds. Using either definition of isotopy it is readily seen that isotopy is an equivalence relation. This is more transparent using Koecher's formulation: Let rj be an isotopy of H onto H I , C an isotopy of H' onto H" so q and are bijective linear and we have bijective linear maps q* : H' -, H , <* : H" -+ H' such that (5.10.4) holds and U[(,O = CUalCx, a' E H'. Then q-' is an isotopy of H' onto H since (5.10.4) implies that U,-I(,O = q - l U , ~ ( r l * ) - l for a' in H . We observe also that the v* in (5.10.4) is unique and (q-I)' = ( q * ) - l . Next we have for a E H .

<

Hence
(5.10.6)

where p E F* is called the multzplzer of the norm similarity. Since q H ( l ) = 1, p = nHt(q1). We remark also that if T/ is norm similarity of H onto HI and E is an extension field of F then the linear extension q of rj to H E is a norm similarity of H E onto H&. Since the degree of special Jordan algebra coincides with the degree of homogeneity of its reduced norm function, it

5.10. Isotopy and Norm Similarity

243

follows that algebras that are norm similar have the same degree. If we take E = F(C1.. . . , Sm) where the (, are indeterminates and m = [ H : F ] then we ( 7 7 ~ )= pnH F ( s ) ( 5 ) for generic -c. obtalrl The first important relation between norm similarity and isotopy is that isotopy implies norm similarity. This is proved in Jacobson [68], p. 242 for char F # 2 and by hIcCrimmon in [75]in the general case. Since the proof is simpler and less computational if char F # 2 and this is the only case we shall consider in the applications to involutions in simple algebras, we shall reproduce here only the proof for char F # 2. we require the following Lemma 5.10.7. Let F be an algebraically closed field of characteristic and let c be invertible i n F [ c ] .Then c has a square root in F [ c ] .

#2

We omit the elementary proof and remark that 5.10.7 does not hold if char F = 2. We can now prove for char F # 2. Theorem 5.10.8.Any isotopy of H onto H' is a norm similarity.

Proof. We first show that if c is invertible in H then n H ( c( a, ) = n ~ ( c ) n ~ ( a E ) ,H .

(5.10.9)

It suffices to prove this for F algebraically closed. By the lemma, c-' has a square root d. Now Ud is an isomorphism of H onto H(') since CTdl = d2 = c-l and qUab = ~ $ ? ( ~ holds b ) for q = U d . This follows since UVdaUcUdb= UdUaUdUcUdb and UdUcUd= UdUd-2Ud= U ~ U T ~=U1. ~Then Ud-1 = u;' is an isomorphism of H(') onto H and since the reduced norm is invariant under isomorphism, n H ( c( a) ) = n H (Ud-Ia ) = n H ( d p 1 ) 2 n( a~) = n H ( d p 2 ) n H ( a= ) n H ( c ) n H ( a )Thus . we have (5.10.9). This relation states that the identity map is a norm similarity of H onto H(')with multiplier n H ( c ) . Since any isotopy is an isomorphism of H onto an isotope HI('') of H' and the reduced norm is invariant under isomorphism. any isotopy is a norm similarity. From now on we shall assume the base field is infinite. which will permit us to use the differential calculus of rational maps. In particular we can define a trace bilinear form on any special Jordan algebra H by

Here Abf for a rational function f is the rational function a --, A: f (see Section 5.4). Since A?Abf = A! Aaf (5.4.6), t H( a ,b) is symmetric and it is clearly bilinear. Now we shall define H to be separable if t ~ , () is nondegenerate. It is clear from this definition that H is separable if and only if H E is separable for any extension field E I F . Now let H = M n ( F ) + . We shall show that t H ( a ,b) = t r ab. As is well known (and easy to see), t r ab is non-degenerate. Hence it will follow that t ~ ( ab ), is non-degenerate for H = -M',(F)'. First we calculate AknH for any

244

V. Simple Algebras with Involution

b and invertible a. W e recall t,hat nH( a ) = det a for H = hf',( F ) or M n ( F ) + . T h e n AknH is t h e coefficient o f t i n det(a + tb) = (det a ) ( d e t ( 1 + t c ) ) , c = a I b . Now r n , , ~ ( t )= det(t1 - c ) = tn - tr(c)tn-I . . .. Hence

+ c) = t-n t r ( ~ ) t - ( ~ -+' ) t c ) = 1 t r ( c ) t+ . . . .

det(tflldet(1 -

I t follows and

-

-

n;nH= (det a)tr(a-lb)

(5.10.11)

A: log nH = t r ( a P 1 b )

(5.10.12)

for any b and invertible a. W e now denote t h e identity m a p b y x (as is usual i n analysis) and write x P 1 for t h e rational m a p a --, a-I i n H where H is any special Jordan algebra u ~ i t hambient associative algebra A . W e have t h e relation .z.z.-l = 1 i n A . DiEerentiating this we obtain

and since

Akx = b we obtain t h e important formula

for any b and any invertible a i n any special Jordan algebra. I f we apply this and (5.10.12) ure obtain

for a , b and invertible c i n M n ( F ) . Putting c = I we obtain

T h i s implies b y a base and t,he non-degeneracy o f t H ( a ,b) for H = hfn(F). field extension argument Theorem 5.10.15. (i) Let A be central simple associative. Then A+ is separable. (ii) Let A be simple with involution of second kind. Then H ( A , J ) is separable.

JVe consider next t h e special Jordan algebra H ( A . J ) where A is central simple and J is a n involution and we assume char F # 2. Let K ( A , J ) b e t h e space o f J-skew elements. T h e n A = H ( A , J ) @ K ( A . J ) . Let h E H ( A , J ) , s E K ( A ,J ) . T h e n I t ~ ( hs ), = t r ( h s ) = t r - ( h s + s h ) = 0 (5.10.17) 2 since s.h = i ( s h h s ) E K ( A , J ) . T h u s H ( A . J ) and K ( A . J ) are orthogonal subspaces i n A relative t o t h e trace bilinear form t A ( , ). I t follows that H ( A . J ) and K ( A . J ) are non-degenerate relative t o t h e restriction o f t A ( . )

+

5.10. Isotopy and Norm Similarity

245

to these subspaces. Now if J is of orthogonal type then n ~ ( h =) n ~ ( h for ) h E H = H ( A , J ) and if J is of symplectic type then n H ( h ) 2 = n ~ ( h )h . E H. It follows that t H ( h , k ) = t ~ ( hk .) for h, k E H if J is of orthogonal type and t H ( h .k ) = i t A ( h , k ) if J is of symplectic type. In both cases t ~ , () is lion-degenerate. Hence we have Theorem 5.10.18. I f A is central simple over F of characteristic is a71 inuolutiorl i n A, then H ( A , J ) is separable.

# 2 arsd J

Our next objective is to prove the converse of Theorem 5.10.8 for separable special Jordan algebras of characteristic # 2. For this we need some formulas for t H (, ) for any special H of characteristic # 2 like those we derived before for M n ( F ) + for any characteristic. SVe derive first the formula

for invertible a and any b. We may a,ssume F algebraically closed. Let d E H satisfy d 2 = a. Then, by 5.9.4 (i),

n H ( a+ t b ) = n H ( U d ( l

+ (Ud-l b ) t ) = n H ( a ) n ~ (+l U d - Ib ) t ) .

(5.10.20)

On the other hand, as before, the relation n~ ( t l - c ) = t n - t~ (c)tn-l (n the degree of H ) yields n H ( 1 - t c ) = 1 - t ~ ( c ) t . . .. Hence

+

+

+

n ~ ( 1 (Ud-1b)t)= 1 + t ~ ( U d - ~ b ) t. . .

+ .. .

(5.10.21)

and hence, by (5.10.20) we have A:~H =n

~ ( a ) t ~ ( Ub )~) .- l

(5.10.22)

+

Siiice char F # 2. U,b = 2n.(a.b) - a2.b (where a.b = (ab b a ) ) and direct ) t~(a.(b.c)) calculation shows that (a.b).c - a.(b.c) = +[cab]so t ~ ( ( a . b ) . c= by Corollary 5.5.15. It follows that

Then, by (5.10.22), we obtain the required relation (5.10.19). We note also that this implies that A : ~ H =t

H ( a # .b) ~

(5.10.24)

which holds by density for all a , b E H . Also by (5.10.19) we have

A: lognH = t H ( a p l . b ) .

(5.10.25)

By (5.10.25). (5.10.13) and the chain rule for differentiation, we have

for arbitrary a , b and invertible c. In particular, taking c = 1 we obtain

246

V. Simple Algebras with Involution

in any special Jordan algebra with char F # 2. Then, by (5.10.26) we have

(5.10.28)

-A:Ab log nH = tH(UC-la.b) for invertible c and arbitrary a. b. We can now prove

Theorem 5.10.29. Let H and H' be special Jordan algebras over a field F of characteristic # 2 . Suppose H' is separable and rj is a n o r m similarity of H onto H ' . T h e n 17 is an isotopy and H is separable.

Proof. It is clear that if a is invertible in H then ria is invertible in H ' . Applying the chain rule of differentiation to (5.10.22) we obtain

for invertible a and similarly from (5.10.13)

Hence

-A:nb l o g n ~ ~ = qt H l ( ~ & ' ~~a b, ) .

(5.10.32)

n H , v = pnH, p E

F*.Hence Ablog nH =

t ~ ~ ( ~ $ ~ =a tH(U;'a, , ~ b ) b).

(5.10.33)

On the other hand, we are given that Ablog n H 1 q and hence

In particular, t H ,( ~ ~ z ' qqb) a , = t H ( a ,b ) which implies that the trace bilinear form on H is non-degenerate and H is separable. Also the non-degeneracy of the forms implies that the linear map : H + H' has an adjoint q* : H' + H relative to the forms, that is. for a E H ' . b E H we have

q x is bijective linear and using this we obt,ain from (5.10.33)

(5.10.35)

t ~ ( ~ < ; b) ' a= , t ~ ( q * U ~ ; ' ~b )a. , Since t~ (

.

) is non-degenerate this implies that U;'

=

v*U;lv

and

Thus q is an isotopy of H onto H ' . In view of the fact that we have (5.10.14) for A+ for any central simple A and this is identical with (5.10.28), and we have separability of A+, the foregoing proof can be carried over verbatim to prove Theorem 5.10.37. Let A and A' be central simple associative algebras. T h e n any n o r m similarity of A+ onto A'+ is an isotopy of A+ onto A'+.

5.11. Special Universal Envelopes

247

5.11. Special Universal Envelopes The concept of the special universal envelope of a special Jordan algebra is the Jordan analogue of the universal enveloping algebra of a Lie algebra. This is given in the following

Definition 5.11.1. Let H be a special Jordan algebra. Then a speczal unzversa1 envelope of H is a pair ( s ( H ) ,a,,) where s ( H ) is an associative algebra and a, is a homomorphism (of Jordan algebras) of H into s ( H ) + such that if a is any homomorphism of H into B f , B associative, then there exists a unique homomorphism C (of associative algebras) such that

is commutativc. The construction of a special universal envelope is also analogous to that of the universal enveloping algebra of a Lie algebra. We begin with the tensor algebra T ( H ) = F @ H @ (H 8 H ) . . . and let K be the ideal in T ( H ) generated by the elements

+

where 1 is the unit of T ( H ) and l H is that of H. Then, if we put s ( H ) = T ( H ) / K . and let a, be the canonical homomorphism a --t a = a K then (s(H)a,) is a special universal envelope of H . First. it is clear that a, is a - homomorphism of H into s ( H ) + since a , ( l ~ )= 7 and a,,(U,b) = abaabli = Uzb = UgU,o,b. Now let a be a homomorphism of H into B f , B associative. By the characterizing property of the tensor algebra we have a unique extension of a to a homomorphism a of T ( H ) into B. The kernel of the extension a contains the elements in (5.11.3), since a is a homomorphism of Jordan algebras. Hence we have the homomorphism ii --, a ( a ) of s ( H ) = T ( H ) / K into B. This is unique since o,(H) generates s ( H ) . The special universal envelope is unique in the strong sense that if (s(H)', 0;) is another then there is a unique isomorphism ( of s ( H ) onto s ( H ) ' such that (a, = a;. Also s ( H ) is generated by a,H. If we apply the commutativity of (5.11.2) in the special case in which B is the ambient algebra A of H and a is the injection of H in A we obtain the

+

248

V. Siiriple Algebras with Involution

injectivity of a,. Hence, when convenient, we can identify H with it,s image in s ( H ) and take a, to be the injection of H in s ( H ) . We proceed to indicate the important properties of s ( H ) that we shall require. We omit the proofs of all but one of these since they are the same as the corresponding ones in the Lie case which are readily available (see; Jacobson [62]or [68]). Pl~nctorlalproperty. If we regard H and H' as subalgebras of s ( H ) + and s ( H 1 ) +then any homomorphism 7 of H into H 1 has a unique extension to a llornornorphism of s ( H ) into s ( H 1 ) . Base field extension. If E is an extension field of the base field F then s ( H E ) = s ( H ) (with ~ injections as the maps of H and H E into s ( H ) and

HE)).

Main involution. There exists a unique involution x, called the main involution; in s ( H ) such that ~h = h for all h E H . Isotopy property. We recall that if c is an invertible element of the associative algebra A then the c-isotope A(') of A is the vector space A with c-multiplication a,/] = acb and unit c-l. Also the map CL' is an isomorphism of A onto A(') (Section 5.10). If H is a subalgebra of A+ and c and c-' E H then H is a subalgebra, H(') of A(')+.The U-operator in H(') is u(') = UU,. \Ve have noted also t,hat d E H is invert,ible in H if and orlly if it is invertible in H ( ' ) and we can form H ( ' ) ( ~ The ) . U-operator for this special .Jordan algebra is U(')U;') = UU,UdU, = UUUcd.Hence H ( ' ) ( ~= ) H("cd);which is another aspect of the transitivity of isotopy. Also if we choose d = c-2 then Ucd = 1 so H ( C ) ( C - ~ ) = H . Thus H is the c-2-isotope of ~ ( ' 1 Another . remark that we shall need is that if q is a homomorphism of H into H' and c is invertible in H then c1 = qc is invertible in H' and q is a liomomorphisrn of H(') into H 1 ( " ) . TVe now have the following result on special universal envelopes of isotopes that was first not,ed by McCrir~lmon(see Jacobson [76],p. 119).

Proposition 5.11.4. Let s ( H ) (with injection m.ap) be a spec,ial u.niversal

envelope for the special Jordan algebra H and let c be an invertible element of H . Then ( s ( H ) c, L ) is a special universal envelope for ~ ( ' 1 . Proof. Let a be a homomorphism of H(') into B f where B is an associative algebra. Then a is a homomorphism of H = H(') into B + ( ~ C - '=) B("-')+ and hencc a can be extended to a homomorphism of s ( H ) into B ( ~ ' - ' ) .NOW ( is also a homomorphism of the associative isotope s ( H ) ( ' ) into B ( ' J - ' )(0,) = B ( ( ~ C - ~ ) ( ~ C ) ( ~ C - ~\N, ) ) . have ac-1 = 1 since c-1 is the unit of H(') and a is a homomorphism of H(') into B+ whose unit is 1. Also

<

" ' and - ~ )()is a homomorphism of s ( H ) ( ' ) into B such Hence B ( ( " ' ~ ~ ) ( ~ ~ )=( B that ( I H = a . We recall that c~ is an isomorphism of s ( H ) ( ' ) onto s ( H ) .

5.11. Special Universal Envelopes

Hence C L I H is a homomorpl~ismof H(') into s ( H ) + . hIoreover, a homomorphisrn of s ( H ) into B such that

249

C' = < c i l is

is commutative. It remains to show that C' is unique. For this it suffices to show that c L ~ ( C = )cH generates s ( H ) . Now cH contains c = cl and cH contains c(&-la) = ac-I for any a E H . Hence the subalgebra of s ( H ) generated by cH contains H and since H generates s ( H ) , cH also generates s ( H ) . We shall now determine the special universal envelopes for the special Jordan algebras H ( A , J ) where ( A ,J ) is a central simple Jordan algebra with involution. Tlle case in which the degree of H = H ( A . J ) is one is trivial since in this case [ H : F ] = 1. For the rest we shall distinguish the cases deg H > 3 and deg H = 2. We consider the former first and we shall prove Theorem 5.11.5 (F.D. and N. Jacobson [49]). Let ( A ,J ) be a central 3. Then A simple algebra ,w,ith involution such that the degree of H ( A , J ) and the injection of H ( A , J ) in A constitute a special universal envelope for H ( A ,J ) .

>

54Te shall base the proof on the following theorem that does not require finite dimensionality. Theorem 5.11.6. Let ( D ,j ) be an associative algebra with involution ( j : d --i d) and let J he the standard in~iol~etion (d,,) -. (d,, ) i n hl,,( D ). Assume 12 > 3. Then iVIn(D) and the injection constitute a special universa,l enuelope for H ( M n( D ) J ) .

.

-

Proof. The conclusion is equivalent to: If A = iZln(D) and 0 is a homomorphism of the special Jordan algebra H ( A . J ) into B+, B associative, then has a unique extension to a homomorphism of A into B. If d E D and i # j then d [ i j ] deij deji E H ( A , J ) . Also, if 6 E H ( D ,j ) : then 6 [ i i ]= Seii E H and every element of H ( A , J ) has the form CT 6i[ii] x i < , di,[ i j ] 6, , E H ( D ,j), dtj E D . The elements e,i, 1 5 i 5 n, are orthogonal idempotents in H ( A , J ) . As noted before the associative conditions for t,his are

+

which are equivalent to the Jordan condition

+

250

V. Simple Algebras with Involution

(see p. 198). We also have the following important Jordan relations on the elements l [ i j ]= e,, e,i, i # j :

+

(5.11.9b)

err o 1 [ i j ]= 1 [ i j ] l [ i j ]o l [ j k ]= l [ i k ] , i ,j , k

#

(5.11.9~)

eii o l [ j k ]= 0 = l [ i j ]o l [ k ! ] , i ,j , k , ! Finally we note that

x

#.

(5.11.9d)

n

t,, =

1.

(5.11.10)

1

Now let a be a homomorphism of H ( A ,J ) into B f and put g,, = gij = a l [ i j ] = g j i , i # j . Then applying a to the foregoing relations we obtain for i f j, (5.11.11) 9%; = gii, u g z z g j 3= 0 = ~ 1 0% gjj g?. = gii $ 9 . . 27. 33 (5.11.12~)

1

Now put f.. -

- Sii, f i j = giigijgjj, i

z2

Then

:f

= fti: f i i f j j = 0 ,

# j.

(5.11.14)

i#j

and f i j = giigijgjj = (gij - gzJg,i)gJj (by 5.11.12b) = g,,gj,. grzgiJ.Then for i ,j , k #,

Also f,:, fke = 0 if j f

2.7

Similarly, f i j =

# k and any i and !and

.f..32 -

2 giigijgjigzi = gtzgi3gzi = g,i(gii

= g 2.2.

- f -

. ,,'

+ gj3)gii

Thus the f,j satisfy the multiplication table for matrix units:

where no restriction is placed on the indices. Also we have

5.11. Special Universal Envelopes

C

fit

=

251

1.

It is classical a,nd elementary (see e.g. BAI. Exercise 9, p. 9 7 ) that B = iCI,(C) where C is the centralizer of the f,j in B and B = lZ4n(C)means that a.ny b E B can be written in one and only one way in the form C c t j f,,, cr, E C . For d E D and i # j we have d [ i j ] = Uem",e33d[ij]. Hence a d [ i j ] = LTfss.,,a d j j j ] = f , i ( a d [ i j ] )f,, f,, ( a d [ i j ]f,,) which implies that

+

for linear maps q and vjl of D into C . In particular, if d = 1 then ul[zJ]= ?, 923= f Z Z g V f j 3 .f31g23.f2L = f,, f,, (k)y 5.11.14). Hence

+

+

Let d l , d2 E D and i 3j: X: obtain

# then d l [ i j ]o d z [ j k ] = d l d z [ i k ] .Applying u we ( ~ , , d l ) ( g k d 2 )= 77zkdld2.

(5.11.19)

Putting da = 1 we obtain v z j d l = ,rlrlzkdland put,ting dl = 1 me obtain rljkdz = v l k d a It follows that qij = vkc for any (i.j ) ; ( k ,1) such t'hat i # j and k # !. Call this map q . Then (5.1 1.18) and (5.11.19) state that 17 is a homornorphis~n of D into C and hence ( d t j ) --t (qdz,) is a homomorphism of M',(D) into B = iZ.fn(C). We show next that C is an extension of a . By (5.11.17) a d [ i j ] = ( d [ i j ] for any d E D and any i # j . If the involution j is tracic in the sense that any 6 E H ( D , j ) has the form d d then u ( 6 e 2 i ) r < ( S e t ) follows from he,, = U e z ( l [ i j o] d [ i j ] )if 6 = d + d. Since the elements d [ i j ] and 6ezi span H ( D : j ) Itre have a = I H ( n / f n ( D ) ,J ) . Since any j is tracic if char F # 2 this shows that is an extension of a in the characteristic # 2 case. To prove this in general we require an argument due to Martindale. First, we show that if d E D : 6 E H ( D , j ) and i # j then

<

+

<

<

IfT? have fie,,

a(6det,

0

(dr,,

+ae,,)

+ d p J Z )= fide,, + &e,, and hence a(Se,,) o a(de,, + de,,) + dSe,,). Then

=

= ((Sde,,

Since U,%%(fieii) = Seii and ae,i = f i i , a ( 6 e i i ) E C f i i . Then the left hand side of (5.11.21) reduces to a ( 6 e i i ) ( v d )f z g (7d)f j i a ( 6 e t , ) . Hence u ( 6 e i i ) ( q d )f i j = v ( 6 d )fij which implies (5.11.20). We can now calculate

+

252

V. Simple Algebras with Involution

This completes the proof that ( is an extension of CT.It remains to prove the uniqueness of this extension. This follows from the easily proved fact that H ( h f n ( D ) )generates AJ,,(D) if n 2.

>

Theorem 5.11.6 remains valid if standard involutions are replaced by canonical ones. This is clear since H ( M n ( D ) :J ) for ,J-canonical is isotopic to H ( A ' f n ( D ) :i ( ? ) J ) where -/ = diag(y1, . . . , y,,). ri E ( H ( D ) j, ) . Hence we can invoke Proposition 5.11.4. Theorem 5.11.6 for canonical involutions was first proved by Jacobson and Rickart ( [ 5 2 ]wit,h ) the restrict,ion that j is tracic. This restriction was removed by Martindale who proved a more general result (Martindale [67],Jacobson [68]and [76]). We can now give the

Proof of Theorem 5.11.5. By the base field extension property of the special universal envelope. it suffices to prove the result for F algebraically closed. I11 this case we have seen that the following is the list of simple algebras with involution.

1. ( A f n( F ) 9-Vfn( F ) ' , E ) , E the exchange involution ( a . b) --i (b. a ) . 2. ( M n ( F ) ,t ) , t the transpose involution. 3 (A12,(F), t s ) . t s the symplectic i~lvolution(see p. 192).

-

The first of these is isomorphic to (&fn(F@ F ) ,J ) where J is the standard involution defined by j : ( a . P ) --i (0. a ) . For, it is readily verified that tlie map ( ( a r 3& .) ) ( ( c Y ~ )( P, ~i j ) ) is an isomorphism of (AJn(F @ F),J ) onto (-Vfn(F)0 IW:,(F)',E).The algebras in 2. are matrix algebras with standard illvolution and vie have seen (p. 233) that t,his is also the case for the algebras in 3. In all the cases listed above deg H ( A , J ) = n. Hence Theorem 5.11.5 in t,he algebraically closed case follows from Theorem 5.11.6. We now consider tlie central simple algebras with irlvolution ( A . J ) such that deg H ( A . J ) = 2. Since it costs nothing to do so we consider more generally any special Jordan algebra H of degree 2. For any a E H we have

and hence for a , b E H we have

+

where nH( a ,b) = nH( a b) - n H ( a ) - n H (b) is symmetric and bilinear in a and b. It follows from these equations that

U,b = aba = n a ( a ) bf t ~ ( a ) t ~ ( b-)t a~ ( b ) n ~ $ ( n ) l -n ~ ( a b)a. , Since n ~ ( X -1a ) = X 2 hence

-

(5.11.24)

t ~ ( a ) X + n ~ ( a ) , n ~ ( l=+l a+) t ~ ( a+) n ~ ( a and ) n ~ ( l , a=) t ~ ( a ) .

(5.11.25)

5.11. Special Universal Envelopes

253

Now put

(5.11.26)

Si = t ( a ) l - a.

Then = t ( 1 )1 - 1 = 2.1 - 1 = 1 and ii = t ( a )1 - rL = a. Using these forrnulas we ran write the U-operator in the more compact form

The norm form n l f is a quadratic forrn such that n H ( l ) = 0.If Q is a quadratic form on a vector space V then we shall call a point 1 E V such that Q ( 1 ) = I a base point for Q. Evidently if Q # 0 and Q ( c ) = /3 # 0 then 5 - l Q has the base point v. \Ire shall now define a Clifford algebra for a quadratic form with base point and we shall show that this a.lgebra for ( n , HI, ) is a special universal envelope for the special Jordan algebra H of degree two. As uslial we write & ( a ,b) = Q ( a b) - Q ( a ) - Q ( b ) . \fie now define T ( a ) = &(a. I ) , a = T ( a ) l - a. Then T ( 1 ) = Q ( 1 , l ) = 2 and hence as above 7 = 1 and E = a. T;CTecan now define the Clifford algebra of Q with base point 1 as follows.

+

Definition 5.11.28. Let V be a vector space equipped with a quadratic form with base point ( Q . 1 ) . Then me define the Clzflord algebra C ( V ,Q , 1) = T ( V ) / K where T ( V ) is the tensor algebra of V and K is the ideal in T ( V ) generated by the elements

where I T is the unit of T ( V ) (and hence of F )

+

We write a = a K , a E T ( V ) . so I is the unit of C ( V .Q . 1) and we have a2 - T ( a ) g + Q ( a )1 = 0 , u E V . The map a --i a is universal for these properties in the sense that if o is a linear map of V into an algebra A such that 01 = 1 and ( 0 ~- T) (~a ) ~ aQ ( a )1 = 0 t,hen there is a unique homomorphism C ( V ,Q , I ) into A such that a * u a , n E V . Sfre shall call ( Q , 1) pure if V = F 1 $ Vo where T ( c o )= 0 for ~0 E Vo. This is always the case if char F # 2 since in this case T ( 1 ) = Q ( 1 , l ) # 0 and me can take V , = ( F l ) l ,the orthogonal complement of F1 relative to Q ( , ) .

+

Proposition 5.11.30. If ( Q ,I ) is pure then C ( V ,Q , 1) E C(K,- Q ) the usual Clifsord algebra of the quadratic form -Q / Vo on Vo. Proof. As usual we regard V = Fl%Vo as a subspace of C(\To/o. -Q) and we have a; = - Q ( a o ) l for a0 E T/o. Also since T ( a o )= O , T ( a ) = 2n: for a = a 1 f a,-,. Then a2 = a21 2aao a; = a 21 2aao - Q(a0)1 = -a2 1 2aa - Q(a0)1 = T ( a ) u- Q ( a )1. Hence we have a homomorphism of C ( V ,Q , 1) into C(V0.- Q ) such that a --i a. a E V . On the other hand we have = - Q ( a O ) l . Hence by the basic property of the usual Clifford algebra, we have a homomorphism of C(Vo.- Q ) into C ( V .Q , 1) such that a0 --i no,ao E Vo. Checking on the

+

+

+

+

Sii

254

V. Sirrlple Algebras with Involution

srts of generators Vo for C(V0,- Q ) and V for C ( V ,Q , 1). we see that the two homomorphis~nsare inverses. Hence both are isomorphisms. 0 It remains to consider the case in which ( Q , 1) is rlot pure (so char F = 2 and T $ 0 ) . Choose d E V so that T ( d ) = 1. Then V = Fd 8 Vo whcre b$ = { b E V 1 T ( v ) = 0). Also me can writ,e Vo = F l @ W where Tit' is any complement of F1 in Vo. Let C(T/V,Q ) be the Clifford algebra of Q = -Q oil W . \Ve have a derivation D' of the tensor algebra T ( W )such that D'b = b+Q(b, d ) l , h E I!V. Since this maps b ~ b - Q ( b ) l > b E IY into 0 it stabilizes the ideal L ge~leratcdby the elements b % b - Q ( b ) l . Hence it defines a derivation D in C(L;II.Q ) = T ( W ) / L such that

TVe have D2b = Db and since D 2 is a derivation (in characteristic 2), D 2 = D . \Ve now form the differential polynomial algebra C ( W ,Q) [t.D ] of polynomials co clt c2t2 . . . where

+ +

+

+

+

+

+

This relation implies ct2 t2c = Dc so c(t2 t ) = ( t 2 t ) c . Since t 2 t also cornrrlutes with t , t 2 t and t 2 t Q ( d ) l are central. Hence the left ideal I generated by t 2 t Q ( d ) l is two-sided. Let B = C ( W ,Q ) [ t :D ] / I . It is readily seen that the image of C ( W ,Q ) i11 t,he canonical homomorphism of C(1V.Q ) [ t D , ] into B is inject,ive so we may idcnt,ify C ( W ,Q ) with its image. It is also easy to see that B = C(1K Q ) C(IV,Q ) u where u = t I and if co clu = 0 for c, E C ( W .Q ) then co = 0 = el. Hence [ B : F] = 2 [ C ( WQ ) : Fj : lye can now prove

+ + +

+ +

+

+

Proposition 5.11.33. C ( V ,Q , 1)

+

"B .

+ + + +

Proof. Let a E V . Then a = ad pl b where a , R E F , b E I.V and T ( n ) = a ,& ( a ) = cu"(d) f12 Q ( b ) a8 a Q ( d ,b). We define a map a of V into B by cr(ad+pl+b) = a u + p l + b . (5.11.34)

+ +

+

+ Q ( a ) l = 0 since a=)( a~u + /3l+ b)2 = a 2 u 2+ 0 2 1 + b2 + a ( u b + bu)

Then cr is linear. 01 = 1 and ( c r ~ ) T ~ (a)oa

( ~

+

Hence we have a homomorphism of C ( V ,Q , 1) into B such that ad+ Bl b --i cuu+pl + b If b E W , b2 = Q ( b ) l .Hence we have a homomorphism of C ( W ,Q ) into C ( V .Q , 1 ) such that b --, b. If a E V then G 2 + ~ ( a ) a + Q ( a )=l 0 and hence if a , b E V then ~ i o = b ~ ( a ) b + ~ ( b ) l i + Qb( )al.. In particular, if b E W and d is

5.11. Special Universal Envelopes

255

as above then b d f db = b + Q(b, d ) l . This implies that the inner derivation in C(V, Q , 1) determined by d maps the subalgebra generated by the 6, b E W, into itself. LIoreover, we have a homomorphism of C(W, Q) [t,dl into C(V, Q, 1) such that b --, b if b E W and t -+ d. Since dZ + d + Q(d)l = 0, the ideal I defining B as C(W, Q)[t,d ] / I is mapped into 0 by this homomorphism. Hence we have a homomorphism of B into C(V, Q, 1) such that b -+ 6, b E W. d. Again, checking on generators shows that the homomorphisrn we and 1~ defined of C(V, Q , 1) into B is the inverse of the one we defined of B into C(V. Q, I ) . Thus B C(V, Q, 1).

-

"

The two cases we considered imply Corollary 5.11.35. If [V : F ] = n then [C(V,Q,1) : F ] = 2"-l.

Proof. We recall that the dimensionality of a Clifford algebra defined by a quadratic form on an n-dimensional vector space is 2n. If (Q, I ) is pure then, by 5.11.30, C(V, Q , 1) Z C(Vo,-Q) where [Vo: F ] = n - 1. Hence [C(V,Q, 1) : F ] = 2n-1 in this case. If (Q, 1) is not pure, by 5.11.33, C(V,Q, 1) B. Moreover, [B : F ] = 2[C(W,Q) : F ] and [W : F] = n - 2. Hence [C(V,Q, 1) : F ] = 2.2n-2 = 2"-'.

"

The foregoing result is all the information we require on the algebras C(V, Q , 1). Further results can be found in Jacobson and McCrimmon [71]. We return now to the special Jordan algebras of degree 2. Theorem 5.11.36. Let H be a special Jordan algebra of degree 2. Then C ( H , nH, 1) and the canonical map a --t a of H into C ( H , n H , 1) (which is injectzve) constitute a special universal envelope for H.

Proof. We have noted the following universality property of C ( H , nH, 1): If a is a linear map of H into an associative algebra A such that al = 1and ( ~ a-) ~ t H (a.)aa nH ( a ) l = 0, a E H , then there exists a unique homomorphism of C ( H , nH, 1) into A such that a --t a a , a E H . Our result will follow from this if we show that a linear map a of H int,o A is a homomorphism into A+ if and ~ n H ( a ) l = 0. The necessity of these only if a 1 = 1 and ( 0 ~-) tH(a)(aa) conditions is clear and the sufficiency follows since we can deduce from those conditions the homomorphism condition U,,ab = a(U,b) as on p. 198.

+

+

We can now prove the following extension of Theorem 5.11.5 to the degree 2 case. Theorem 5.11.37. Let (A, J ) be a central simple algebra with involution such that deg H ( A , J ) = 2. Then A and the injection of H(A, J ) in A constitute a special universal envelope for H(A, J) in all cases except that in which J is of symplectic type.

Proof. Since C ( H , n ~I ) , H = H ( A , J) is a special universal envelope for H we have a homomorphism of C ( H , nH, 1) into A such that a --, a , a E H . We

256

V. Sirnple Algebras with

Involution

now compare dimensionalities of A and C ( H , nH.1). It is readily seen that in all ca5es except that in which J is of symplectic type we have equality and hence isomorphism. In the symplectic case [H(A,J ) : F]= 6. [A : F]= 16 and [ C ( H .n H . 1) : F ] = 25 = 32. Hence A C ( H , nH.1) in this case. For the applications it is useful to formulate the special case of Theorems 5.11.5 and 5.11.35 in which A is central simple and defines the algebra with irivolution (A e A', E ) in the following way:

Theorem 5.11.38. Let A be a central simple algebra. T h e n A 8 A0 and the m a p a,, : a --i ( a ,a ) constitute a special universal algebra for A+. This follows from the indicated results and the fact that a isomorphism of special Jordan algebras.

--,

( a , a ) is a n

5.12. Applications t o Norm Similarities Throllghout this section we assume the base field is infinite. We prove first the following result, the second part of which for A = & ( F ) is a classical result due to Frobenius [1897].

-

Theorem 5.12.1. 1. T w o central simple algebras A1 and A2 have similar reduced n,orms ( n ~ , n A , ) if and only if they are eith,er isomorphic or antiisomorphic. 2. T h e reduced n o r m similarities of a central simple A are the m,aps x --i axb (5.12.2)

where a and b are invertible elements of A, unless [AI2 = 1, i n which case they are the maps (5.12.2) and the maps

where a and b are invertible and x -i x" is any anti-automorphism of A. (Existe.rzce of such a n an,ti-automorphism is clear since [AI2 = 1.) Proof. 1. If A1 and Az are norm similar then so are A: and A t . Then, by Theorem 5.10.37, A: and A t are isotopic. Then, by Proposition 5.11.4, s(A?) E s(A$) and hence. by Theorem 5.11.36, Al a?.A: r A2 @A!.Since the components A, and A: are simple it follows that either A1 2 A2 or Al E A!. Hence either A1 and A2 are isomorphic or they are anti-isomorphic. 2. It is clear that the maps (5.12.2) are norm similarities and the same is t>rueof t,he maps (5.12.3) if x * x* is a n anti-automorphism. Now let 7 be any similarity of the reduced norm and let 71 = c. Then a = c i 1 7 is a norm similarity, hence a n isotopy of A' such that 01 = 1. Then u is a n automorphism of A+ and hence we have a unique autonlorphism of A @ A0 sllch that

<

5.12. Applications t o Norm Similarities

257

where 5,,: a --i ( a , a ) , is commnutative. Now C either has the form ( a , b) --, (
1% consider next isomorphisms and norm similarities for algebras H ( A . J ) with A simple and we prove first Theorem 5.12.4. Assume char F # 2 and let (Ai, J,), i = 1 , 2 be central simple such, that A, is simple and exclude (A,,J,) with J, of symplectic type zf deg H ( A , , J,) = 2. Then, H(A1: J1) 2 H(A2, J2) ++ (AI, 51) (A2.52).

Proof. The hypotheses imply that Aiand the injection map constitute a special universal envelope for H ( A i , J i ) (Theorems 5.11.5 and 5.11.35). Hence if 77 is a n isomorphism of H(A1: J1) onto H(A2, J 2 ) then 17 has a unique extension t o a n isomorphism of Al onto Aa. Now 77 is an isolnorphism of (Al. J1)onto (A2.,J2).For: if a l E H(A1, J1) then r,Jlal = Val = .J2r/al and since H(A1, J1) generates A l , q J l a l = J277al holds for every a1 E Al. Thus H(A1: J1) r H(A2, J2) + (A1 , J1) (A2. J z ) . The converse is clear.

"

Before proceeding we shall need to record some general facts about structure groups of special Jordan algebras. If 77 E Str H then 77 is an isomorphism of H onto the isotope H(') where c = 77(1)~'. Also if s ( H ) and the injectlion r of H in s ( H ) is a special universal envelope for H then ( s ( H ) ,c L ) is a special universal envelope for ~ ( ' 1 .Hence we have a unique automorphism of s ( H ) such that

<

258

V. Sirnple Algebras with Involution

is commutative. Hence for a E H .

a (7a)c = Applying the main involution .ir to this relation we obtain ~ ( = c - ' ( ~ ( ~ a ) )= c cP1((a)c = c-l(C7ia)c (since H C H ( s ( H ) ,7i). This relation shows that for the anti-automorphisms n( and (n we have ~ ( =a ic-l
Hence va E H. Next let a. b E H . Then

Hence IT,, = qU,q* where v* = v-~U,-I which is the condition that v E Str H. We remark that the element c such that (5, J ) = i, is determined up to a multiplier y which is an invertible element of C n H , C the center of A. Replacing c by r - l c , 7 invertible in C n H replaces 7 by m = (y-le)zl< and this is the only alteration that can be made for the given automorphism C of

A.

The condition (C; J ) = i, for an automorpllism C of A is automatically satisfied if ( is inner. Let = i,-I, g E A. Then for any a E A,

<

5.12. hpplicatio~lst o Norm Similarities

250

Thus (<. J ) = i , holds with c = i - l g - l ( J y ) - l and the corresponding element of Str H is the map c;'ig-1 = y(Jg),qig-1 which is

Since Str H is a subgroup of the group of norm similarities we have the following

Proposition 5.12.9. Let ( A , ,J) be an algebra with inz~olutionand let C be an automorpllism of A such that (C,J ) = < J < - l J = i,. Then 7 = (:il(H is a n o r m similarity for H = H ( A , J ) . Moreouer. for any invertible g E A an.d any inuertible y E H n C , C the center of A , the map (5.12.8) is a norrrl, si~milarity.

\ItScan now prove Theorem 5.12.10. Same hypotheses as i n Theorem 5.12.4. then: 1. H ( A 1 , J l ) and H ( A 2 ,J 2 ) have similar reduced norm, form,s if and only if there ezists an invertible cn i n H ( A z ,J z ) such that ( A 1 ,J1) ( A a ,i,, J z ) . 2. If A is central sim,ple (so J is of first kind) then the n o r m similarities of H ( A , J ) are the maps a --i ygag", g' = Jg (5.12.11)

where E F+ and g is inuertible i n A . 3. If the center of A is a quadratzc extension K of F and J is an involution of second kind i n A the n the n o r m similarities of H ( A . J ) are ,yrrst the m,aps (5.12.11) un,less A has arc outer au.tomorphism. I n the latter case if i f is such a n automorphism then there exists a n invertible c E H ( A , J ) such that i f J i f - l J = i c and then th,e norm, similar.ities are either of the form (5.12.11) or of the jorm

where 7 E F* and g is an inuertible element o,f A . Proof. 1. If 7 is a similarity of n H , onto n,H,, H , = H ( A i . J,), then 11 is an isotopy (Theorem 5.10.29), hence an isornorpl-lism of H I onto an isot,ope H;"') of H z . Now H;'" ) " ( A z , i,, J 2 ) (scc p.239). Hence H ( A 1 , ,71) H ( A p , i,, .J2). Tlien ( A l ,J l ) E ( A l ,i,, J 2 ) by Theorem 5.12.4. 2. By Proposit,ion 5.12.9, the maps in (5.12.11) are norm similarities. By Theorerns 5.11.5 and 5.11.35, ( A ,i) is a special universal envelope for H ( A , J ) . Rloreover, J is the main invohtjon T in A since J = T on the set of generators H ( A , J ) . Also, by the Skolern-Noether theorem, the automorphisms of A are all inner. It follows that any 77 E Str H ( A , J ) has the form (5.12.11). Since. by Theorern 5.10.29: any norm similarity is contained in Str H ( A , J ) . these have the forin (5.12.11). 3. Let i f be an outer autornorphism of A . Then p I K = J / K and hence p J I - ~ c l is J an automorphism of A I K . Then p J ~ - l J= i,, c E A and hence

"

V. Simple Algebras with Involution

260

c - l ( p a ) E Str H ( A , J ) . Also if p' is a second outer automorphism then K = 1 K so p' = i g - l p , g E A . The rest of the argument is identical witll that of 2. n

--i

1 '

I n Theorems 5.12.4 and 5.12.10 we have excluded the ( A ,J ) in which A is central simple of degree 4 and J is of symplectic type. We now consider what happens in this case. Here the degree of H ( A , J ) is 2. [ H : F ] = 6 for H = H ( A , J ) and the norm form n,H on H is a quadratic form. Let Ho be the 5-dimensional subspace of H of elements a such that t H ( a ) = 0. By Theorem 5.11.35 and t,he isomorphism of C ( H ,n H . 1 ) with the Clifford algebra C ( H o .- n H ) , C ( H o ,- n ~ and ) the injection map of H = F1+ Ho into C ( H o ,- n H ) constitute a special universal envelope for H. Since [ C ( H o -. n ~ ): F ] = 25 = 32 and [ A : F] = 16. A is not a special universal envelope of H ill the present case. However, there is something that can be salvaged in this as we proceed t o show. T;Cie recall Albert's theorem that if A is a central division algebra of degree 4 then A = A1 8 ~ 4 where 2 Ai is a quaternion algebra. Evidently this is the case also if A = 1 % f 2 ( 3 ) : D of degree 2 or if A = M d ( F ) .Hence for any involutorial central sirrlple A of degree 4 we have a factorization A = A 1' @ F A 2where Ai is a quaternion algebra. Let Ji denote the standard in~olut~ion a, --, u, = t ( a , )l -a, in A, a,nd let A: = { a , E Ai I t ( a , ) = 0). Since t ( a , ) l = ai Si,, Aj is the subspace of Ji-skew elements. Since [ H ( A , .J i ) :F] = 1, J , is of synlplectic type. Hence if u , is a n invertible element of Aj then J,,% = i U z J iis of orthogonal type. Then J = J1,, 'E Jz is of sylnplectic type in A = A 1 gE. A2. Now put

+

L(a1. a z ) = ul 15 a2

+

(5.12.13)

2 ~ 1 @ ~ ~1 1

for (1, E A:. Then ( ( a l .a z ) E H = H ( A , J ) and since { [ ( a l .a z ) I a, E A / , ) is a 6-tiimensional subspace of A',

Since a l . nl E A ; , t ( u l a 1 ) = tila1 +lLlal= u l a l

+ O I U l = a1u1+ u l a l . Also

--

where n ( u l a 1 ) = ( u l a l ) ( u l a 2 )is the reduccd norm in A l . Then

[ ( a l ,~

+ u l a l @ 1)' = n ( u l ) n ( a a )+ u l a l 8 az + E a2 + (ulal)' E 1 = r ~ ( u l ) n ( a z+) t(u1al)ul 8 a2 + t ( u l a l ) u l a l 2 1 n ( u l a l )

2 = )(

~~ 1 a2 %

?LI~IUI

-

= t ( u l a l ) [ ( a la, z ) - n ( u l ) ( n ( a l )- n ( a Z ) ) .

It follows that

n ~ ( l ( a ai z, ) ) = n ( u i ) ( n ( u l ) n(aa)). \TTe now give the following

5.12. Applications to Norm Similarities

261

Definition 5.12.17. Let A be an involutional central simple algebra of degree 4 over F . A = X F A 2a factorization of A as tensor product of quaternion algebras and let A/, be the space of elements of trace 0 in A,. Then the quadratic form on A: $ A', defined by

is called an Albert quadratoc form on A. 5tTecan now prove for char F

#2

Theorem 5.12.19. (i) Any two Albert quadralic forms for an involutorial central simple algebra of degree 4 are similar. (iz) TWOin.riolutoria1 central simple algebras of degree 4 are isomorphic if arzd only if their Albert forms are similar. (iiz) If Q is an Albert quadratic form for A then the index of A is 4: 2 or 1 according as the Witt index of Q is O,1, or 3.

Proof. (i) Consider the map (al,a2) --i t ( a l , a n ) , a zE A:: of A: € A', into H ( A ,J ) where J is the symplectic involution defined above. This ma,p is bi. n(u1).Hence this is a jective linear and n H ( t ( a la, 2 ) )= p(n(a1)- n ( ~ ) p) = similarity of nH with an Albert form on A: @ A;. Since any t,wo norm forms defined by involutions of sylnplectic type are similar, (i) follows. (ii) Slippose A and B are involutorial central simple algebras of degree 4 ~ i t similar h Alhert forms. Then we have symplectic type involutions J and K in A and B respectively such t>hatH ( A . J ) and H ( B ,K ) are norm similar. Then, by Theorems 5.10.18 and 5.10.29, H ( A ,J ) and H ( B . K ) arc isotopic. Hence, by Proposition 5.11.4, the special universal envelopes s, (H( A ,J ) ) and s , ( H ( B , K ) ) are isomorphic. Since H ( A ,J ) and H ( B ,K ) are of degree 2; s,, ( H ( A ,J ) ) and s, ( H ( B K , ) ) are 32-dimensional Clifford algebras C and C'. Now C (C') is either simple or it is a direct sum two isomorphic simple ideals (Theorem 4.12, BAII) and we have a hornornorphism of the special universal envelope C onto A. Since [ C : F] = 32 and [A : F] = 16 it follows that C is a direct sum of two sirnple ideals and these are isomorphic to A. Similarly C' is a direct surn of two simple ideals isomorphic to B . Since C and C' are isomorphic it follows t,hat A and B are isomorphic. The converse is clear. (iii) Suppose first that A = h f ~ ( F@) A> We can take u1 =

(11)

above to get ,J1,, ( u l ) = t a l . Viewing A = I I I ~ ( A ~we ) )have . H = H ( A ,J ) =

(

)

+

Ia.=BtF.atA2.

+

for^=

(E i ).

it is easy to check that

x2'-(a p)z ( a p - n ( a ) )1 = 0, so nH (z)'= ap - n(a).Since the diagonal matrices form a hyperbolic space, we see Witt index n~ = I + Witt index n. If Az = AJz ( F ). then n = det has Witt index 2, while n is anisotropic for A2 = D, a division algebra. It follows that for A of index 1 or 2, Q has Witt index 3 or 1 respectively. Finally. suppose A is a division algebra and let J be a simplectic

262

V. Simple Algebras with Involutiorl

t,ype involution on A. We claim that n~ for H = H ( A , J ) is anisotropic, which will imply that the Albert forms have Witt index 0. Otherwise, we have a n a # 0 in H such that n ~ ( a = ) 0. Then a is not invertible in F [ a ] (see p. 205) contrary t o the hypothesis that A is a division algebra. Part (iii) of the foregoing theorem is a n extension of Theorem 2.10.9. Also revcrsing the order we had in Chapter 2 we can derive from 5.12.19 (iii) for char F # 2 the

Corollary 5.12.20 (Albert-Sah). Let A,: i = 1:2. be a q ~ ~ a t e r n l odivision n . is not a diuisio~nalgebra zf and only if the A, algebra. Then A = A1 3 ~A2 contain isomorphic quadratic subfields (cf. 2.10.,?). Proof. The condition is sufficient since the tensor product of isomorphic fields is never a field. Now suppose A1 @ F A2 is not a division algebra. Then, by 5.12.19 (iii) the Albert form associated with the given factorization of A as A1 @ p A2 is isotropic. Hence there exist a , E A: such that ( a l , a z ) # 0 and ,(al) = ,,(a2). Then ai # 0, a: = -n)(ai)l. Hence F(a1) F ( a 2 ) .

"

Remarks. 1. The analogue of Corollary 5.12.20 for algebras of degree three fails, for, it has been shown by Tignol and Wadsworth in [go] that there exist central simple division algebras Ai of degree 3 such that Al 8~Az is not a division algebra but A1 and A2 do not have isomorphic cubic subfields. 2. In [90] P. Mammone and D.B. Shapiro have given a version of 5.12.19 that is valid for characteristic two.

5.13 The Jordan Algebra H ( A , J ) In this section we shall generalize the results of 3.12 on elements of ra.nk 1 of central simple algebras to the Jordan algebra H ( A , J) where (A, J ) is a central simple associative algebra with involution. We consider first sorne basic Jordan algebra concepts. For the sake of simplicity we assume F infinite of characteristic # 2. (The case of characteristic 2 is t,reated in Jacobson [85].) A subspace B of a Jordan algebra H is called a n inner ideal if 11nb = Uha E B for all b E B and a E H . If b E H then UhH is an inner ideal since Ur;,>,H = UhUo,UbHC UhH. This is called the principal inner ideal det,ermined by b. In particular, we have the Peirce inner ideal U,H determined by the ideinpotent e and the Peirce decomposition H = LTeH @ H@ U I p e H which follows froin H = U I H = U,+(l-,lH. If H is an associative algebra (and hence a special Jordan algebra) we have the two-sided Peirce decomposition H = e H e @ e H ( 1 - e) 6 (1 - e)He a (1 e ) H ( l - e) arid LTeH= eHe. Ue,l-,H = e H ( 1 - e ) + ( 1 - e ) H e , U l - , H = (1 - e ) H ( l - e ) . Ideals in a Jordan algebra a,re defined as for arbitrary non-associative algebras. It is readily seen ill the charact,eristic # 2 case that a subspace B is an ideal if and only if U,B C B for all u in H. This is clear since U, = 2L; - La2 and 2L, = U,,J where 2L, = a,,bl f a , . The Jordan algebra -

5.13 The Jordan Algebra H ( A , J )

263

H is called simple if it contains no ideals # 0 , H . A Jordan algebra is called a division algebra if every non-zero element is invertible. The Jordan algebra D+ is a division algebra if D is an associative division algebra and H ( D , J ) is a division algebra if D is an associative divisiori algebra. A Jordan division algebra contains no proper ideals. In 5.1 we determined the simple algebras with ivolution ( A ,J ) . We showed that these are either of the form B @ B0 where B is simple with the exchange involution cp : ( b l ,b2) --i (b2,bl) or of t,he form M m ( D ) where ( D ,j ) is either a division algebra or is iVf2(E),where E is a field, and j is a --, (tra)l2 - a. In the division algebra case, J is a canonical involution based on j. In the case in which B = 1bf2(F),J can be taken as standard. In the case in which A = B @ B0 where the involution is the exchange map, B = M,(E) where E is a division algebra. Then A = AJv,(D) where D is E $ Eo and j is the standard involution based on the exchange map in E 9EO. This is similar to the form which we obtained for A simple. The pair ( D ,j ) is called the coefic,ient algebra of ( A ,J ) . It is clear that the coefficient algebra of ( A ,J ) is semi-simple in all cases. Theorem 5.13.1. If ( A ,J ) zs central simple the Jordan algebra H ( A . J ) is szmple. Proof. It suffices to assume that the base field F is algebraically closed since (A, d ) remains central simple on extensions of the base field to its algebraic closure and if H ( A , J ) has a non-trivial ideal, then so does its extension. The involution J is standard in the algebraically closed case. For algebraically closed F the possibilities for ( D .j ) are ( F .I F

/

F ) ; (n/f2(F).(1

"

--,

(tr a)12 - a , ( F CE F, ( a ,b)) --i (b,a ) ) .

Thus in all cases H ( D ,j ) F . Hence if m = 1, H ( h f m ( D ) J; ) r F is simple. If in > 1. the elements of A have the form a = Cai,e,, and those of H ( A . J ) have the form CaijeZ3where aii = tizi and ai, = # j. If a is in an ideal B of H ( A , J ) then so is a,,e,, = Uetta and a,,e,, +aiJe3i = Ue,,.e,,a, i # j. Also for c,:, arbitrary (aiae,, TLzjeji)0 (cZjei, ci, e,i) = (a,,cij c,jti,j)e,, ( a t 2~ 1 3 + E,pzj)ej, We wish to show that if aij # 0, then cij can be chosen so that (a,,&, + cLja,,) # 0. This is trivial in the first and last cases above, and in the second case it follows from the non-degeneracy of the trace bilinear form on AJ2(F). If B # 0 and a # 0 in B then either there exists an i such that a,ie,i # 0 in which case B 3 e,,: or there exist i and j , i # j , such that ( a i j e i j+ a i j e j , ) # 0 in B. Case 1. B 3 e,i # 0 , then for any j # i , e , , o ( e Z j+ e j i ) = ( e t j + e J i ) E B. Hence there exists a c,j such that ( c , ~ cij)eii (ci, C.27. . ) e33. . # 0. Then e j j E B. Hence 1 = eii C j f ie,, E B and so B = A. Case 2. (ai,ei, ai,e,i) # 0 is in B. Then there exists a cZj such that (aiJeij TLi3e3,)0 (ctJeij cijejz) # 0. Then eii E B. So we have a reduction to Case 1.

aji,i

+

+

+ + +

+

+

+

+

+

+

V. Simple Algebras with Involution

263

Definition 5.13.2 An element b of a Jordan algebra H is called of rank 1 if U5H = Fb # 0 and H is said to be reduced if H is spanned by rank 1 elements, that is. H = F p ( H ) where p ( H ) is the set of rank 1 elements. Clearly, ab E p ( H ) if b E p ( H ) and a E F " . Hence H is reduced if and only if every element of H is a sun1 of rank 1 elements. F p ( H ) is an ideal since for b E p ( H ) we have for x E H . ?(x) E F such that U 5 z = y (.r)b. Then

Hence T/,b E F p ( H ) .Hence if H is simple and contains rank 1 elements then H is reduced. In particular, by Theorem 5.13.1 this is the case for H = H ( A . J ) where ( A .J ) is central simple. A11 idempotent element e of a Jordan algebra H is called a dzvzszon rdempotent if the Jordan algebra (U,H, U,e) is a division algebra with unit e. LVe have

Proposition 5.13.4 (McCrimmon [ 8 5 ] ) . If e is a division idempotent in a redu.ced Jordan algebra H then e E p ( H ) .

Proof. Any element of U e ( H ) has the forrn CLTec,where c, E p ( H ) . By (5.13.3) every d, = U,C, E p ( H ) and since d, is in the division algebra U,H. Uds(U,H) = UeH if d, # 0. On the other hand, Uds(U,H) C Fd,. Hence U,H = Fd, if d, # 0 so U,H is one dimensional. Then UeH = F F and e E P(H). IVc now define a composition algebra with involution ( D ,j ) / F if 1. it has no absolute zero divisers, that is, elements z # 0 such that zaz = 0 for all a in D and 2. for ariy a in D , ati E F where a = j ( a ) . The linear span Z of absolute zero divisors in an algebra D is an ideal. Hence Z = 0 if D is semi-simple. The composition algebras with ir~volution are knowli. They are

(1) D = F . (2) A quadratic composition algebra (either a separable quadratic extension field E / F or a direct sun1 of two copies of F,j the a~itornorphism# l E of E). (3) D a quaternion algebra, j the standard involution in D. This is proved in Jacobson [81]pp. 6.5f. \Ve now have

Theorem 5.13.5. If (A, J ) is cen,tral simple and H ( A , J ) is reduced then the coeficient algebra ( D ,j ) is a composition algebra.

Proof. By passing t,o an isotope (see Section 5.10) we may assume t,he involution J is standard in all cases. Since e l l is a division idempotent in a reduced

3.13 The Jordan Algebra H ( A , J)

265

Jordan algebra, it is of rank I and so acell E e l l H e l l = F e l l . Thus, a a E F and ( D , 3 ) is a composition algebra. The converses, that the algebras ( I ) , (2) and (3) are composition algebras and that if ( D ,j ) is a composition algebra then H ( A , J ) is reduced, are clcar. If el and ez are id em potent,^ in a Jordan algebra H, the associativ~conditions ele2 = 0 = ezel are ecluivalent to the Jordan conditions el o ea = 0, CT,,e2 = 0. This is readily seen. If H = H ( A , J) where (A, J ) is central simple. is reduced then (A, J) = M m ( D ,j) where m is the degree of H ( A . J) a,nd (D.j ) is a co~npositionalgebra. Then H ( A , J) contains the pair-wise othogonal elements eii, i = 1,. . . , m,. Let ( u l , . . . , IL,,) be a base for H = H ( A , J ) where (A, J ) is central simple . . . , En be indeterminates. ViTeshall now derive a system S = { f ) and let of homogencous polynomials f in F[(1:. . . , [,I such that if E is any extension field of F and a?E E;1 5 i 5 n:then a = Cn,u, is of rank 1 if and only if f (a)= f ( a l , . . ,a,)= 0 for all f E S. Consider the element x = C<,u, of HF(o, F(E)= F(E1,. . . ,En). Suppose

Since U,z,,l = U,,]

.,,%

and

U

= 2Uu2we have

$% ,,

where

7L


):'%kt<: + )hlJku
(5.13.10)

2 <3

Then UxukE F ( < ) x for all uk if and only if

Then if E is any extension field of F and a = Ca,uiE HE, Uauk E En! I k 5 n, if and only if

<

for all where

V. Simple Algebras with Involution

266

Since Ua?lkE E a . 1 5 k 5 n , implies U,HE c E a , it is clear that the equations f J k e ( a )= 0 hold if and only if a E p ( H E ) . The polynomials (5.13.12) define a projectzue varzety of rank 1 elements of H , denoted by RH is an F-variety. \VP shall show that RH is irreducible. For this purpose we introduce the operator

Ba,b = 1 -

+ UaUb

(5.13.14)

b - U,b2 (V, = U,,l).

(5.13.17)

va,b

Then we have the identities

B0,bBb.a = 1 - V,

+ Uc where c = a

0

See Jacobson [81],p. 3.37. These can be verified by expressing the terms in terms of left, and right multiplications. For example,

Lrt e be an idcmpotent, H = UrH @Ue,1-,H @ Ul-,H. the Peirce decomposition relative to e. If a E Ue,l-,H then

Hence B,,, is invertible with B,: = B-,,,. h!toreover similarly, B e , , is invertible with B;: = B e . -a. Va,,e = U,,,e = Ve~,a= V,a = a. Hence

If a. b E Ur,l-,H then B,,,b = b-V0,,b+U,Ueb = b-Ua.be and Ua,beE Ul-,H. Thus

B,,,b = b - U,,be. Li,,be E Ul-,H

Finally, wc note that U,+,e and c E Ul-,H then

=

Uee

(5.13.20)

+ Ue,,e + U,e = e. Hence. if e is of rank 1

e+c€p(H)@c=O

(5.13.22)

We can now prove

Lemma 5.13.23 (Faulkner). If e is a rank 1 idempotent in H and H = LTeH€3 Ue,l-eH @ U1-,H, the corresponding Peirce decomposition, then u = e a c, where a E CI,,l-,H, c E Ul-,H, is in p ( H ) if and only if c = U,e in which case u = B-,,,e.

+ +

5.13 The Jordan ,4lgebra H ( A . J )

267

Pmof. Suppose first that u E p ( H ) . Since B , ~ , is invertible. (5.13.16) shows B,,,uE p ( H ) if B,,,u#O. Now B,,,IL= B , , , ( e + a + c ) = e - u + u u e + a LJu.,e+c (by (5.13.19),(5.13.21))= e+cl,cl = U,e-lJa,,e+c E Ul-,H. Then c' = U,e-2li,e+c = 0 by (5.13.22).Thus c = U,e and by (5.13.6) 2~ = B-,,,e. Conversely suppose c = U,e. Then B -,. e = e + a + U-,e = e + a + U,e = u and since e E p ( H ) ,'uE p ( H ) . Theorem 5.13.24. If ( A ;J) is central simple then H = H ( A . ,J) is reduced if and only if H contains nz pair-wise orthogonal idempote~ztswhere m is the degree of H .

Proof. We have seen that t,he condition is necessary. To prove it,s sufficiency. suppose e l ! . . . ; em are pair-wise ort,hogonal ideinpot,ents in H . Suppose H is not reduced. Then U,_H contains an element b $! Fe,. Let v ( X ) be the rniniillunl polynomial of b so deg u ( X ) > 1. Choose a l ; 0 2 : . . . a,-1 distinct ate,+ and not roots of v ( X ) .Then the minimum polynomial p(X) of a = b is v ( X ) - a , ) . This has degree > m and hence the reduced minimum polynomial of a is of degree > m. This cont,radicts that deg H = m.

xyp1

n:-'(X

Let G = Str H . By p. 241, r j E G if and only if q is bijective linear and there exists a bijective linear transformation v' of H such that U,(,) = vU,q* for all a E H. Putting a = 1 we obtain U,(l) = rjq* so rj* = rj-lUq(l). Since v - ~- (det 77)-17# where q# is the adjoint of the linear transformation, the

~ ( ~ are ) . equivalent t'o conditions on q become (det v)U,(,) = r j ~ , r ~ # ~Thesc a set of polynomial equations on the coordinat,es of the matrix of 7 relative to a base for H I F . It follows that G is a linear algebraic group. Let Go be the component of 1 of G . Thc map a --i U, is a continuous (in the Zariski topology) map of the open subset of H defined by N ( n ) # 0 into Str H. Since the image contains 1 it follows that every U, for a invertible is contained in Go If e is an idempotent and a E I ~ , . ~ - , Hthen we have seen that B,,, is invertible. Then B,,, E Str H by (5.13.16). The argument used for U, implies E G o . We can apply these results to prove that B,,, Theorem 5.13.25. Let G o be the component of 1 of G = Str H . Th,en G stabilizes p ( H ) and Go acts tm,nsitively o n p ( H ).

Proof. It is clear from the condition U,(,) = rlUaq* for 7 E Str H that pH is stabilized b y StrH. Now H contains an idempotent of rank 1) e # 0. To prove the second assertion me have to show that Goe = p ( H ) . Since Go contains the set F* of noii-zero scalar multiplications and every B,,,r+ E p ( H ) for n E U,,l-,H it follows from Faulkner's lemma that Goe contains every u = a e a c E p ( H ) such that a E F * ; a E &,l-,H) c E Ul-,H. Now let v be any element of p ( H ) . bTeclaim that there exists a k and invertible a l : .. . . ak such that U,, . . . U,,v $! U,,l-,H Ul-,H. Otherwise, by Zariski density, for all k and a l . . . . ak E H , U,, . . . U,,v E U,~l-,H+Ul-,H. Then v generates a

+ +

+

268

V. Simple Algebras with Involution

proper ideal in H contrary to simplicity. Hence we have invertible a , such that CJcLl. . . UuLz~ in Goe. Since U,, . . . U,, E Go, 21 E Goe and p[H) = Goe. This result implies that we have an induced transitive action of Go on

Rlr Now it is well known that if we have a transitive action of an irreducible group on a n algebraic variety V and the action is defined by polynomial maps then V is irreducible (see Jacobson [63] p. 33). Hence 5.13.24 implies Theorem 5.13.26. If (A. J) is central sim,ple then, RH for H = H ( A , J) is irreducible. ifre shall consider next the structure of R H for H reduced (equivalent,ly, by Tlleorerli 5.13.5). H has coefficient algebra a c~mposit~ion algebra.

Theorem 5.13.27. Let H = H ( A , J) be reduced. Then RH is birationally equizlalent over F to a projective space P I T , V = Fe+U,.l-,H; e an idempotent of rank 1.

+

Proof. Let 0 be the set of points F ( e a ) in P V a E U,,l-,H. Then 0 is ari open set in PV.Next let 0' be the set of points F ( e a c) where n ill Ue,l-,H and c in U1-,H. By Faulkner's lemma. 0' is a n open set in RH and the map F ( e a ) --, F ( e a U,e) is bijective regular since in terms of a suitable choice of homogeneous coordinates it is given by homogeneous quadratic polynoinials with coefficier~tsin F . Its inverse is a projection map. Since RH and P V are irreducible they are biratioilally equivalent (see e.g. Hartshorne [77]p. 24 f ) .

+

+ +

+ +

Corollary 5.13.28. F ( R ) o zs ratzorlal oller F of trnnscendancy degree [U, l - , H : F].

Definition 5.13.29. An extension field EIF is called a red,ucing field for H if IIE is reduced. E is called a gen,eric reducing field if it has the property that a n extension E'IF is a reducing field for H if and only if there exists a n F-place on E to E'. Since HE is simple, it is reduced if and only if it contains a n element of rank 1. It is clear that this is the case if and only if RH has a n E-rational point. Hence if we have

Theorem 5.13.30. E/F is a reducing field for H if and only if R H has a 7 ~ E ra.tional point. One of the main theorems on reducing fields of H = H ( A . J ) is the following

5.13 The Jordan Algebra H ( A . J)

269

Theorem 5.13.31. (1) The field of rational functions I?(%&)/ is a generic reducing field ,for H . (2) F ( R H l o is ra,tionul over F if and only if H is reduced. (3) F ( R H ) o is a regular extension of F and
Theorem 5.13.32. H = H ( A , J ) llas a finite dimensional separable reduci~ay field.

Proqf. Let rrL be the degree of H. Let ~ r l. ,. . , u,,be a basis for H : s = Cy [,u, where the tiare indeterminates, and let d ( x ) = d ( & ; . . . , E n ) be the discriminant of m,(X), the minimum polynomial of x. We claim that d ( z ) # 0. To see this let F be an algebraically closed extension. Then H p is reduced and hence H F contains rn orthogonal idenipot,ent elements c, # 0. Let a = ELn=, aie,, where the ai are distinct elements of F. Then the minimum polynomial p,(A) of a is 17(A - a,) so m,(X) = p,,(A). Hence if n = C@,u,,P, E F : then d ( y l : .. . : 3,) # 0. So d ( x ) # 0. (Cf. Lemma (5.3.14); p. 200). Since F is E F so that if c = C y , u , then d ( c ) # 0. Theri infinite we car1 choose m , ( X ) = pc(A) has distinct roots in F . Now let F, be the separable algebraic closure of F and consider c as element of Hp,. Since p,(A) has distinct roots in Fs: the subalgebra Fs [c]of HF,- and hence H F , contains ~ rn non-zero distinct orthogonal idempotents. This implies that Hps is reduced. Now let f = C p i u , be a reduced element of HFs and put E = F(cpl,. . . $ 9 , ) . Then E is finitedimensional separable over F and H E contains the reduced element f . Hence H E is reduced. Theorem 5.13.33. Let H = H ( A , J ) be as usual. Suppose E is a separable extension field of F such that [E : F ] = m = deg H and E f is isomorphic to a subalyebrr~of H . Then E is a reducir~gfield for H .

Proof. LW have E = F ( p ) where p has rnirlirnurn polynomial f ( A ) of degree m with distinct roots. Also H contains an element a with minimum polynomial f ( A ) = I T ( X - p,) where the p, are in the algebraic closure F of F and pl = p. Put f Z ( A ) = f ( X ) / ( A- p,)f'(pz) = -PJ)/ - PI).

n ( ~ U(P%

I#,

J#,

Then ,f,( A )f , ( A ) r O(mod f ( A ) )and C f, ( A ) - 1 is a polynomial of degree m - 1 which is 0 for the rn-values p,. Hence C f , ( A ) = 1. Then substituting X = a we obtain e, = f,(a) which satisfies the equations ez = e, and e,e, = 0. Hence H ~ .P,) ( is~reduced ~ by (5.13.24). Since el E H F ( p )H . F ( p )is reduced. So E is a reducing field for H.

V. Simple Algebras

270

with Involution

W e consider next the norm hyper-surfaces S H where H is as before. I f the base field is algebraically closed then n H ( x )is either the determinant o f (EL,) where <2,7 are indet>erminates,or o f the generic syinmet,ric matrix (C,,) where Ci, = <,,, or is the P f (2s)where z = ( & ) , Cz, = 0. CzJ = - C j 2 . In these three cases it is classical that n ~ ( zis) an irreducible polynomial. Hence ( n H ( x ) is ) ) ) a domain. the ideal o f S H and the coordinate algebra F [ S H ]= F I C ] / ( n I < ( x is Hence we can define the field o f rational functions (llomogeneous o f degree 0 ) o f S H . UTeshall call this field the projective norm field of S H and denote it as F ( S H ) ~W. e shall show that i f H is reduced then S H is birationally equivalent t o a projective space PV where V is n-dimensional, n = dim H . W c prove first some general results on reduced norms o f a Jordan algebra H o f degree m. I f a and b art elements o f a Jordan algebra H of degree 7n we have r~(+ a Xb) =

1n,(a, b)XZ

T h e n n o ( a ,b) = n ( a ) .n,(a, b) = n ( b ) . and in general, n,(a, b ) = n ,,-, (b.a ) is a homogeneous polynomial funtion o f degree nl-z in a and of degree z in b. T h e - rl ( a )A7+l . .+(- 1)" T , ( a ) = reduced minimum polynomial ma( A ) )= rz(X1 - a ) - a. Hence r L ( a )= n t ( l .a ) . (5.13.35)

+.

Now let H = H ( A , J ) where A = & ( D ) and ( D ,j ) is a composition algebra with involution and J is a canonical involution based on j . Let {ei, / i, j = 1 , . . . r r ~ )be the matrix units in the representation o f A as h f n ( D ) . Tllen the e,, E H . Let e = e l l . W e claim the reduced minimum polynomial m , , ~ f ( X )= Ampl(X - 1 ) . Consider the subalgebra H f = LTeH U1-,H = F e U I p , H o f H . Since H f is a subalgcbra o f II deg H' 5 deg H . O n the ot)her hand, a = Ca,ei, E H f and i f the a , are distinct the minimum polynomial o f a is 17(A- a , ) which is o f degree m. It follows that deg H' = m. Since H' = F e @ U1-,H and these are ideals in H ' , i f b = bl bn where bl E Fe and bz E Ul-,H, m b , ~ ) ( A=) m b , . ~ , ( A ) m b , , u , - e( A~ ) . Applying this ) m , , ~ t ( A= ) ( A - 1)A7"-I. t o bl = e and b2 = 0 we obtain r n , , , ~ ( X= IVe can now prove

+

+

+

Theorem 5.13.36. (1) Th,e norm hypersr~rfaceS H where H = H ( A , J ) a,n,d ( A ,J ) is central simple and H ,is reduced is birationally equ,ivalent over F to a projective space PV where [V : F ] = [H : F ] - 1. (2) The field of rational furlct,~or~s F ( S H ) Ois rational m:er F of tmnscendency degree [ H : F ] - 2.

Proof. (Faulkner) ( I ) \Ve know that A = ilI,(D) where ( D ,3 ) is a composition algebra, m is the degree o f A and J is a canonical involution based on J . I f r is as before then me H ( A )= ( A - l ) A m - l . Hence, by (5.13.35). n , ( I . e ) = 0 i f > 2. Let c be an invertible element o f H and consider the isotope H(')

5.13 The .Jordan Algebra H ( A , J)

271

whose unit is c-l. Since e is rank one in H it is also of rank one in ~ ( " 1Since . n(')(.x)= n(c)n(x) we have n(c)rz,(c-l, e) = 0 if i 2 2. Hence

holds for all b. by Zariski density. If a E H , a = cue b where b E V = Ul-,H

+

+ la ,,-,

H. Hence

Let 0 be the open subset of PV defined by n(b)nl (b. e) # 0 and O1 the subset of S I f defined by n(b) # 0. Then we have the bijective F map q : F b --i F a . a = -n(b)e+nl(b, e)b of 0 into 0 1 . This is regular with regular inverse. Since SHand PVE are irreducible they are birationallv equivalent over F. Evidently. [V : F] = [H : F] - 1. (2) This is an immediate consequence of (1).

Theorem 5.13.39. If H = H(A, J) where (A, J ) is central sim,ple then F ( S H ) ~is a generic reducing field for H and H is reduced if and only if F(SH)O is ratiorlal over F .

Proof. Let x = CT (,u,where the [, are indeterniinates and the uz constitute a base for H . The reduced rninirnum polynomial nl, (A) = n(A1-x) is irreducible in F[
>

272

V. Simple Algebras with Involution

Theorem 5.13.40. If D is a cen,tml division algebra then K is a splitting field (generic splitting field) for M,(D) if and only if K is a reducing (generic reducing field) for the Jordan algebra (M,,,( D ) + .

Proof. We have the isomorphisin a --, ( a , u ) of Al,,(D)+ onto H(lI,f,(D @ D o . J ) . If K is a splitting field for l\lm(D), AI,(D)K = hf,(I<) and hence i\l,(~)& = ATn ( K ) + and H(114, ( D @ D o )K : J ) = H (&&In ( K 3 KO);J ) . Thus K is a reducing field for H(il'f,, ( D $ D o ), J ) and hence for &Im ( D ) + . Conversely. if K is a reducing field for Al,(D)+, then I \ [ ~ ( D K ) n/l,(D;), J ) is J ) where C is a composition algebra reduced and so has the form H(n/l",,(C), over K . Since the only composition algebra over K that is not simple is K B K , it follows that A[,,(D)K = A)I,(K) so K is a splitting field for Af,(D). The statement on generic reducing fields and generic splitting fields follows from what me have proved and the definitions of the generic objects.

+

Corollary 5.13.41. Let D be a centra.1 division algebra. T h e n the fields of ,ration,al fun,ctions F(Rnr,(u)+) and F(Snr,(D)+) are generic splitting ,fields for AJm(D). T h e field of rational functions o n the Brauer-Seueri variety V(12T),(D)) is a generic reducing field for h I m ( D ) + .

Proof. The first statement follows from 5.13.40 and 3.13.31 and 5.13.39. The second follows from 5.13.40 and 3.8.7. Next let ( D ,j ) be a central division algebra with involution j of second kind. Thus D has center C which is a quadrat,ic field extension of the base field F and j is an involution in D such that jjC # l c . Let J be the standard involution in A&(D) based on j. We say that an extension field K I F is a ( D ) K = AJn(CK). Since C is scparable splitting field of (;2,1,(D), J ) if CK is either a direct sum of t,wo copies of K or is a field according as K coilt,ains a subfield isoinorphic to K or not. Accordingly, we say that K is a type I or type I1 splitting field. It is clear also that C K and the linear extension of jlC t,o CK is a conlposition algebra over K so H ( A / l , ( C K )J, ) is reduced as algebra over K . Since * / I , ( D ) C K = M n ( C K ) ,K is a reducing field for H ( M , (D), J ) . Conversely, suppose K is a reducing field for H(ll.lm,( D ), J ) . Then ( H ( A J ( D )J, ) K = H ( A ! f n ( C 1J) ;) where ( C ' ,j ) is a composition algebra over K. Since the degree of H(I& ( D ) ,J)l< = deg H ( M , ( D ) , J ) and [H(lLf,(D), J ) K : K ] = [ H ( M m , ( DJ, ) : F ] ,these conditions and the determination of the composition algebras (p. 264) shows that ( C ' :j ) is either K @KO with j as the exchange map or C' is a q1iadrat)icfield over K with j as the automorphism # lcl of C ' . Accordingly, we say that K is a kype I or type I1 reducing field. Hence we have Theorem 5.13.42. If ( D :j ) is a central division algebra with involution of second kind then a n extension field K I F is a splitting field of type I or type IT o,f M,(D) if K is a reducing field of H(n/l,(D), J ) of type I or type II, J the standard involution based o n j .

5.13 The Jordan Algebra H ( A ,J)

273

If we define generic splitting fields of type I and type I1 of M,,(D) and generic reducing field of H ( M r , ( D ) , J ) in the obvious way we can deduce from 5.13.42 the corresponding statement for the generic objects.

Theorem 5.13.43. T h e field F ( V A ) ~A , = iZ&(D), of rational functions o n the Brauer-Severi variety and the field F(S*)o of rational functions o n the n o r m hypersurface are generic type 11 reducing fields for H ( M m( D ) ,J ) .

Proof. By Theorem 3.8.7, F ( V A ) ois a generic splitting field for A and hence a generic reducing field for H . Moreover. F(VA)ois regular over F which implies that F ( V A ) o contains no finite dimensional extension of F f F . In particular. F ( V A A ) , Jis a type I1 extension of F . The statements for F ( S A ) ofollow in the same way from Theorem 3.11.11. Next suppose ( D ,j ) is a central division algebra with involution of orthogonal type. Let A = M m ( D ) and J the standard involution in A based on j. Then J is of orthogonal type since the degree of H ( A , J ) is md where d is t,he degree of D and [ H ( A J, ) : F ] = ( m d ) ( m d 1 ) / 2 . If K is a reducing field for H , ( H ( A ,J ) ) K is isotopic to H ( M n ( C ) ,J ' ) where (C,j') is a composition algebra over K and J' is the standard involution based on jl. Then C = K since the other possibilities for composition algebras can be ruled by evaluating the degrees and the dinlensionalities of H ( i V n ( C ) ,J ' ) . It follows that A K is isotopic to iM,(K) and I( is a splitting field for A. Conversely, any splitting field for A is a reducing field for H. In particular, F ( R H ) " and F ( S H ) oare generic split,ting fields for A and F ( V A ) is~ a generic rcducing field for H . An extension field is called a 11s-splitting field for a central simple algebra A if Ah-has index 5 s. This is equivalent to: A K has a splitting field of degree 5 s . We shall nowTsay K I F is a strictly 11s-splitting field if AK has index s. In a similar ma,nner we define it generic 11s-splitting field and generic strictly 11s-splitting field. Now suppose ( D ,j ) is a central division algebra # F with involution of symplectic type. Then the degree d of D is even: say: d = 26. Then the degree of H ( A .j ) is 6 and, by Theorern 5.3.18.3: H contains a separable subfield of degree 6. Let K be any subfield of D of degree 6. We claim that K is a st,rictly 112-splitting field of D . First. K is not a splitting field for D since [ K : F ] < deg D . On the other hand, [ D K : F ] [ K : F ] = [ D : F ] = 46' so [ D: K ~ ] = 46 and [ D K : F ] = 4. Thus D~ is a qua.t,ernion division algebra over K and hence contains a quadratic subfield K 1 / K . Then K' > K a,nd [K' : F ] = 26 = d. Then K 1 is a splitting field for D (1.5.17).It follows that the index of D K is 2. Hence K is a strictly 112-splitting field for D.

+

Theorem 5.13.44. Let ( A ,J ) be a central simple algebra with iwuolution of symplectic type. T h e n there exist strictly 112-splitting fields for A.

Prooj. W7e know that A = kl,,(D) where ( D , j ) is a ccntral division algebra wit11 involution. Tlle possibility that ( D .j ) is of 2nd kind can be ruled out

274

V. Simple Algebras with Involution

since the centers of A and of D is F . If ( D , j ) is of orthogonal type then (A, J ) is of orthogonal type contrary t o hypothesis. Hence ( D , j) is of symplectic type. Then D and hence A has a strictly 112-splitting field. Our main result on involutions of symplectic type is Theorem 5.13.45. Let (A, J ) be a central simple algebra with involution of symplectic type, H = H ( A , J ) . Then a n extension field K I F is a reducing field (generic reducing field) for H if and only if it is a 112-splitting field (generic 112 splitting field) for A. Moreover, if A oo 1 and let K I F be a generic 112splitting field for A is a strictly 112-splitting field.

Pmof. The first statement follows as for involutions of orthogonal type (p. ). Now suppose A oo 1 and let K I F be a generic 112-splitting field. Suppose K is not a strictly 112-splitting field. Then K is a splitting field. By 5.13.44, there exists a strictly 112-splitting field K' for A so A K ~= M m ( D ) where D is a quaternion division algebra over K'. Since K I F is a splitting field and we have a n F-place on K to I<', K ' is a splitting contrary t o AK, = n/J,(D) where D is a division algebra. Since F ( S H ) oand F ( R H ) o are generic reducing fields for H, we have the Corollary 5.13.46. The fields F ( S H ) o and F ( R H ) o are generic 112-splittzng fields for A.

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Index

adjoint (reduced), 26 adjoint (in a Jordan algebra), 207 adjoint identity, 29 Albert quadratic form, 161 Albert’s theorem (splitting of p-algebras), 166 algebra with involution, 186 algebraic derivation, 22 ambient algebra, 197 appear over a field (group), 154 Amitsur-Rowen-Tignol example of degree eight, 229f. Artin-Schreier construction of p-extensions, 158 associated (Brauer) factor sets, 50 associates, 10 atom, 11 bicentralizer theorem, 64 birational equivalence, 117 Brauer algebra, 54 Brauer factor set, 47 Brauer field, 95, 101, 125ff., 130 Brauer group, 41, 51 Brauer group functor, 149 Brauer-Severi variety, 113, 272 c-multiplication, 241 canonical involution, 193 Cayley-Hamilton theorem, 24f. central algebra with involution, 186 characteristic polynomial, 25 Clifford algebra, 199, 253 coefficient algebra, 263 Cohn, Paul M., 38 cohomology group, 103 cohomology set, 104 composite of fields, 89 composition algebra with involution, 264 conjugacy conditions, 46 conjugate, 15 continous cohomology, 92f.

coordinate algebra of a variety, 111 corestriction, 151 crossed homomorphism, 102 crossed product, 57 cyclic algebra (C, σ, γ), 19, 59f. cyclic algebra (α, β), 159 cyclic element, 68 decomposed algebra with involution, 232 degenerate matrix, 181 degree of an algebra, 25 degree of special Jordan algebra, 200 derivation, 1, 205 differential extension (D, δ, d), 23 differential Laurent series, 40 differential polynomial ring, 5 directional derivative, 204 discriminant, 27 discriminant of an orthogonal involution, 234 division idempotent, 264 Euler’s differential equation, 205 exchange involution, 187 exponent of a division algebra, 61 exterior algebra, 107 factorial, 13 Faulkner’s lemma, 266 Faulkner, John R., 110, 270 Fein, Burton, 137 Feit, Walter, 46 form (of a space), 96ff. free composite, 102 Frobenius, Georg, 24 Frobenius algebra, 42 Frobenius endomorphism, 155 Fundamental theorem of arithmetic in a PID, 12 g.c.d. = greatest common divisor, 10 Gauss’s lemma, 25

N. Jacobson, Finite-Dimensional Division Algebras over Fields, c Springer-Verlag Berlin Heidelberg 1996, Corrected 2nd printing 2010 

282

Index

generalized cyclic algebra, 19, 60 generic abelian crossed product, 181 generic element, 24 generic minimum polynomial, 200 generic norm, 200 generic splitting field, 123 generic trace, 200 grading, 63 Grassmannian, 108 Haile, Darrel, 194 Heuser’s theorem, 271 Hilbert Nullstellensatz, 111 Hochschild’s theorem, 156 homogeneity, 53 Huneke, Craig, 146 idealizer, 13 ideal of a variety, 111 index of a division algebra, 60 index of a cohomology class, 107 index of a simple algebra, 30 inflation, 88, 92 inner derivation, 5 inner ideal of a Jordan algebra, 262 inverse in a special Jordan algebra, 236 involution, 186 involutorial algebra, 193 irreducible element, 11 isotope, 241 Jordan algebra, 185 Jordan division algebra, 263 Jordan, Pascual, 197 kind of involution, 190 Koecher, Max, 241 K¨ othe’s theorem, 28 l.c.m. = least common multiple, 10 Laurent series, 37ff. length, 12 Li, Guo, 59 Lie algebra, 185 Lie invariant, 206 linear automorphism, 129 logarithmic derivative, 205 main involution, 199 Martindale III, W.S., 251 matrix units, 250f. McCrimmon, Kevin, 186, 248 Merkurjev-Suslin theorem, 85

multiplier, 242 Noether factor set, 57 Noether, Emmy, 59 Noetherian, 3 non-cyclic division algebra, 79 non-cyclic p-algebra, 183 norm, 25 norm hypersurface, 138 norm similarity, 242 normalized (Noether) factor set, 57 normalized polynomial, 238 Ore-Wedderburn condition, 3 p-algebra, 31, 154 Peirce decomposition, 262 PID = principal ideal domain, 3 place, 123 Pl¨ ucker coordinates, 108 Pl¨ ucker equations, 110 profinite group, 93 projective norm field, 138, 270 projective space, 111 projective variety, 266 q-generating set, 223 quotient division ring, 4 rank 1 element, 141ff. rank 1 element of a Jordan algebra, 264 rational function field, 114 rational map, 117, 205 reduced adjoint, 26 reduced (Brauer) factor set, 48, 54 reduced Jordan algebra, 264 reduced norm, 25 reduced trace, 25 reducing field, 268 regular field extension, 101 restriction homomorphism, 90f. Roquette, Peter, 130 Rowen’s lemma, 71, 215 Rowen, Louis H., 186 Saltman, David J., 85, 144 section of a group (relative to a normal subgroup), 97, 100 Seligman, George S., 50 semi-invariant, 132 semilinear automorphism, 97 semilinear transformation, 86, 91 separable algebra, 28

Index separable field extension, 101 separable special Jordan algebra, 243 sg, sign function, 108 Shapiro’s lemma, 132 signature, 79 similarity, 11 skew polynomial ring, 2 special Jordan algebra, 197 special universal envelope of a special Jordan algebra, 247 Speiser’s theorem, 86, 106 Speiser-Noether theorem, 56 splitting field, 27, 53, 273 standard involution, 192, 210 structure group, 242 Tamagawa, Tsuneo, 152 trace, 25 tracic involution, 251 transcendental derivation, 22

283

transfer homomorphism, 153 twisted Laurent series division ring, 38 twisted polynomial ring, 5 twisted power series ring, 37 two-sided element, 6 two-sided maximal, tsm, 12 type of involution, 190 universal division algebra, 70 unramified Jordan algebra, 200 valuation ring, 123 Wedderburn’s factorization theorem, 67 Wedderburn’s norm condition, 30 Zariski-topology, 25f., 114 zero-divisor in a special Jordan algebra, 236