FiniteDimensional Division Algebras over Fields
Nathan Jacobson
FiniteDimensional Division Algebras over Fields
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FiniteDimensional Division Algebras over Fields
Nathan Jacobson
FiniteDimensional 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 CataloginginPublication Data
Jacobson, Nathan, 19101999 Finitedimensional division algebras over fields / Nathan Jacobson. p. cm. Includes bibliographical references (p. 275280). ISBN 3540570292 (Berlin : hardcover : alk. paper) 1. Division algebras. 2. Fields (Algebra) I. Title. QA247.45.J33 1996 512’.24dc20 9631625 CIP
Corrections of the 1st edition (1996) carried out on behalf of N. Jacobson (deceased) by Prof. P.M. Cohn (UC London, UK) ISBN 9783540570295 eISBN 9783642024290 DOI 10.1007/9783642024290 Springer Heidelberg Dordrecht London New York Mathematics Subject Classification (1991): 13XX, 16XX, 17XX © SpringerVerlag 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 acidfree paper Springer is part of Springer Science+Business Media (www.springer.com)
PREFACE
These algebras determine, by the Sliedderburn Theorem. the semisimple finite dimensional algebras over a field. They lead to the definition of the Brauer group and to certain geometric objects, the BrauerSeveri varieties. Sie shall be interested in these algebras which have an involution. Algebras with involution arose first in the study of the socalled .'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 JeanPierre 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. Skewpolynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Arithmetic in a PID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3. Applications to Skewpolynomial 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. Woncyclic 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 Noncornmutativc Cohomology . . . . . . . . . . . . . . . . . . . . . . . 102 3.4. Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.5. BrauerSeveri Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Table of Contents
viii
3.6. 3.7. 3.8. 3.9. 3.10. 3.11. 3.12. 3.13.
Properties of BrauerSeveri Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 BrauerSeveri 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 . pAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. Noncyclic pAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 aderivation (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 onesided 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. Skewpolynomial 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 aderivation of R , that is, S is additive and for a, b, E R )
6(ab)= (aa)( 6 b )
+ (6a)b
N. Jacobson, FiniteDimensional Division Algebras over Fields , © SpringerVerlag 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) Rmodule (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)
+
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 twosided elements of a ring of differential polynomials.
Theorem 1.1.32 (Amitsur [ 5 7 ] ) . fi) T h e twosided 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
td z={tpe+71tpe'+..+7,td
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. Skewpolynomial Rings
9
Proof. (i) The condition that an element is twosided shows that any inonic twosided 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 = td if char D = 0 and z = t"'flltpe'+. . ietd ~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 ppolynomial f ( A ) = Ape ylApel . . . ?/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 td E Cent R and in the second case if f ( 6 ) = id with 6d = 0 then t p e ?ltpel . . . ?,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 Rmodules 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 nonzero 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 nonzero 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 Rmodule 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 nonzero 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]culand (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 Rn~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 retrare 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 JordanHolder 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 nonzero and nonun%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 (twosided) ideals or equivalently o f twosided 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 twosided element p twos7ded
1.2. .4rithmetic in a PID
13
maximal (t.s.m.) if Rp" is a nonzero 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 twosided 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 twosided 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 twosided 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 twosided 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 antiisomorphism 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 twosided element into irreducibles then the Rmodules 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 antiisomorphic to !VIn(EndRR/Rpi). Since Endfi R/Rp, is antiisomorphic to B,/ Rp, we have R/RpX iVfn,(A),A B,/Rp,.


1.3. Applications t o Skewpolynomial 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 Rmodule 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 Rmodules they are isomorphic as Dmodules. 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.
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. (twosided 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 twosided 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 5conjugate 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 = babl.
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 Skewpolynomial Rings
tr  u
= n I ( t  b,) where the b, are 0conjugates of b and R/R(tr  u ) 121r(Db) ~ ~ h e D r eo = { a E D I Da = bnbI).
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 ppower 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 6conjugate 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,ltrl 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,ltrl)
r (a0
+ a l t + . . . + a,ltr')a
(mod R ( t r  y ~ ) )
I. Skew Polynomials and Division Algebras
20
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 Findependent are Cindependent. 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,onzero elements of Cent R = F [ z ] z, = uIt", 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 6constants 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 ppolynomial 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 yltpel . . . 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 twosided element of R.By Thcorem 1.1.32,we may assume xi* E Cent R,and Cent R = F unless we have a ppolynomial 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 Cmodule 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 JacobsonBourbaki 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 JacobsonBourbaki 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) = pcPl(S~)= 0 SO c p E F . Evidently the subfield F of C is 1dimensional over F and since 6 is an Flinear 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 SkolemNoether 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) = AI 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 Fsubspace 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. e1
(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 nonzero 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 HamiltonCayley 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 HamiltonCayley 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,. . . , 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 ) .
+
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 deth 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 deth 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 ograding 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/tlth 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/tfth 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/tlth roots of unity and they geriera,te the group of d/tlth 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 = clg 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 deth 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 deltth 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 d1
xi'
'
At
=
{ C a j u J aji G E ( ~ ) } 0 t1
(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 deltth 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/tlth root of 1. T4Te note that if we multiply the given a, by a suitable nonzero 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 cgrading. 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 nonzero 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/tlth 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;~'
<
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 peth 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' = (dcdI)z = (c (I((:' c)cll = 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 blAml . . . 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 blarrLI 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 yalyl # 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 = yalyl 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 (Aa;,,) 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 zI f k ( A ) z for all z # 0 . This implies zI 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 zl 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 yay1 # 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 noncyclic 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).
<
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. Noncyclic Division Algebras of Degree Four Albert has given a number of constructions of noncyclic 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 lioncornmutative 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 nonzero 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 twosided irreducible element of R and (tr2 7 ) / (t7 y r l ) . Since yrl = AT,(c), (tc) (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) = (aT21~1)(aT22c1) , . . 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. Noncyclic 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 = bl.
aQ F. Hence b%l
= a2
+ 1 gives u!
= u2
+ u2,
u
=
Lelnlna 2.10.2 suggests a procedure for constructing a lioncyclic 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 nonsquare 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 ,
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 uEuI = 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 pth 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 pth 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 pth power and ( r a ) a P t is a pth 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 pth 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 pth 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 pth power i n W . Proof. Let IV*P be the subgroup of W * of pth 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 pth 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 pth power in Mr. hIoreover, since DU? 1 , M ( 6 j is not a pth 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 Fsubalgebra 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 Fsubalgebra 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 . . . anlA7L1 # 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_lrrl)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 Pnk(Eo,. . . , Enpl) are contained in F. Also since l,, = 0, hl = 0 and hence Pnl([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 pth 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 hoethcr 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 semilinear 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 asemilinear 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 Fsubspace 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 Findependent are Eindependent. That Vo is an Fsubspace 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 Elinear combinations of the y, E Vo. In particular, since u1 = 1; x is an Elinear combination of y, E Vo. Evidently this implies that V = EVo. Kext suppose yl, . . . ; y, E Vo are Findependent. Then the standard Dedekincl independence argument shows that these elements are Eindependent. 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 asemilinear 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
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 Aautomorphisms 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 asemilinear 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 ([(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 nonzero elements of E([)o, and it also contains the submonoid H of nonzero homogeneous polynomials. Tlie submonoid II is factorial. Hence every element of M can be written in the form f(E) = PI( l F e l t + C z > l Fez1 and UleA = 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' = cpl(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' = cpl(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' = (Ule(,)w)t = nl(Ule(,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 Fmodule in which
Regarding K as Fmodule in this way and A as Fmodule 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 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. pAlgebras
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' 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 pth 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 pth 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 palgebras:
Theorem 4.5.7. A n y palgebra 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]).
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 wpel = 7 E F ( u ) . Since [(E,a! 7): F ( u ) ] = p2(e1), 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.
>
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?kla)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
,
o < i , < n,,
A?,... i , E E [ 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 ( [ ~ : . . . ,&).
I and a u E E* such that U =
(
';) is nundegenerate 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 (AmitsurSaltman [ 7 8 ] ) Let . ( E ,a. U, abelian crossed product in wh,ich r > 2 , n , = pf* > 1: 1 5 i nondegenerate. 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 [ & ,. . . : 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 nonzero 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 nonzero 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 nonzero 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 nonzero orthogonal idempotents e, with C e , = 1.
+
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)m2a.
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)m2a#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 = wvwl. 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 nonmaximal 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 (uaul)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 nonzero 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'uI for all d' E D ~ This . implies that D~ is commutative since u  ~(did;)* = dFd;* = ud;d;ul. 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,,)ld,,'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 = 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 nontrivial 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= r1u2uluz = 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,lu2 = &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 ulau2aI au2aIuI # 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 ulau2al 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 nontrivial 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 noncommutative 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 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 = aI. (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 = aI. 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= Ulb1 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 aI 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 ndimensional ~uectorspace with base ( v l : . . . , v,) over a n algebraically closed field F . Let g ( & ; . . . (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 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 twosided. 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 ndimensional 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 ] = 2n1 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 ] = 2.2n2 = 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 antiautomorphism of A. (Existe.rzce of such a n an,tiautomorphism 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 antiisomorphic. 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 antiautomorphism. 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
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 cl. 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,(cl, 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(A1x) 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