Journal of Algebra 239, 356᎐364 Ž2001.
doi:10.1006rjabr.2000.8656, available online at http:rrwww.idealibrary.com on
Exponent Reduction for Projective Schur Algebras1
Eli Aljadeff and Jack Sonn
Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
E-mail: aljadeff@math.technion.ac.il, sonn@math.technion.ac.il
Communicated by Alexander Lubotzky
Received July 10, 2000
In this paper it is proved that the ‘‘exponent reduction property’’ holds for all
projective Schur algebras. This was proved in an earlier paper of the authors for a
special class, the ‘‘radical abelian algebras.’’ The precise statement is as follows: let
A be a projective Schur algebra over a field k and let k Ž ␮ . denote the maximal
cyclotomic extension of k. If m is the exponent of A mk k Ž ␮ ., then k contains a
primitive mth root of unity. One corollary of this result is a negative answer to the
question of whether or not the projective Schur group PSŽ k . is always equal to
Br Ž Lrk ., where L is the composite of the maximal cyclotomic extension of k and
the maximal Kummer extension of k. A second consequence is a proof of the
‘‘Brauer᎐Witt analogue’’ in characteristic p: if charŽ k . s p / 0, then every projective Schur algebra over k is Brauer equivalent to a radical abelian algebra.
䊚 2001 Academic Press
1. INTRODUCTION
The projective Schur group of a field k, denoted by PSŽ k ., is the
subgroup of the Brauer group Br Ž k . generated by Žin fact, consisting of.
all classes that may be represented by a projective Schur algebra A. By
definition, a finite dimensional k-central simple algebra A is projective
Schur if AU , the group of units of A, contains a subgroup ⌫ which spans
A as a k-vector space and is finite modulo the center, that is, kU ⌫rkU is a
finite group. The notions of projective Schur algebras and projective Schur
group are the projective analogues of Schur algebras and the Schur group
of k Ždenoted by SŽ k ... These notions Žof projective Schur algebra and of
projective Schur group. were introduced in 1978 by Lorenz and Opolka w6x.
One of the main points for making this generalization is that PSŽ k .
1
The research was supported by the Fund for the Promotion of Research at the Technion.
356
0021-8693r01 $35.00
Copyright 䊚 2001 by Academic Press
All rights of reproduction in any form reserved.
EXPONENT REDUCTION
357
contains Žby abuse of language. all symbol algebras. More precisely, any
symbol algebra is a projective Schur algebra in an obvious way. ŽIndeed, let
A s Ž a, b . n be the symbol algebra generated by x and y subject to the
relations x n s a g kU , y n s b g kU , yx s ␨n xy, where ␨n g kU is a primitive nth root of unity. It is easy to see that ⌫ s ² x, y : spans A as a
k-vector space and kU ⌫rkU ( ⺪rn⺪ = ⺪rn⺪. In particular, kU ⌫rkU is a
finite group.. Invoking the Merkurev᎐Suslin Theorem, one deduces that if
k contains all roots of unity Žresp. contains the nth roots of unity., then
PSŽ k . s Br Ž k . Žresp. PSŽ k . = Br Ž k . n . where the subscript n denotes
n-torsion. The subgroup PSŽ k . may be large even if roots of unity are not
present in k. Indeed, if k is a number field, then PSŽ k . s Br Ž k . as shown
in w6x. Here one uses the fact that for k a number field, any element in
Br Ž k . is split by a cyclic extension which is contained in a cyclotomic
extension of k. In general, the projective Schur group PSŽ k . is properly
contained in Br Ž k ., e.g., when k is a rational function field k 0 Ž x . over any
number field k 0 . For power series fields k s k 0 ŽŽ x .. Žover a number field
k 0 . the situation depends on the field k 0 . For instance if k 0 is a number
field which is not totally real, then PSŽ k . / Br Ž k .. On the other hand, the
Kronecker᎐Weber Theorem implies that PSŽ⺡ŽŽ x ... s Br Ž⺡ŽŽ x .... As
mentioned above, symbol algebras are ‘‘natural’’ members of PSŽ k . and
clearly, any symbol algebra is split by a Kummer extension of k. In general,
Kummer extensions are not sufficient to split all elements in PSŽ k .. For
instance if k is a number field, Kummer extensions do not split any
element whose order is prime to the number of roots of unity in k
Žrestriction-corestriction argument.. As mentioned earlier, when k is a
number field, every element in PSŽ k . s Br Ž k . is split by a cyclotomic
extension of k. In w2x the following is proved:
THEOREM 1.1. E¨ ery element in PSŽ k . has a splitting field which is the
composite of a cyclotomic extension of k and a Kummer extension of k.
This was a key result used to show that in general PSŽ k . is properly
contained in Br Ž k ., since one shows that Žin general. Br Ž Lrk . is properly
contained in Br Ž k ., where L denotes the composite of all cyclotomic and
Kummer extensions of k.
It is now natural to ask w4, p. 528x: Is PSŽ k . s Br Ž Lrk . in general? In
this paper we show that this is false:
THEOREM 1.2. There are fields k for which PSŽ k . is properly contained in
Br Ž Lrk . where L is the composite of the maximal cyclotomic extension of k
and the maximal Kummer extension of k.
This will follow Žas shown below. from the following theorem which is
the main result of this paper.
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ALJADEFF AND SONN
THEOREM 1.3. Let k Ž ␮ . be the maximal cyclotomic extension of k and let
res : PSŽ k . ª PSŽ k Ž ␮ .. be the restriction map. Then for all ␣ g PSŽ k ., the
exponent expŽresŽ ␣ .. of resŽ ␣ . ‘‘di¨ ides the number of roots of unity in k.’’
More precisely, if m s expŽresŽ ␣ .., then k contains a primiti¨ e mth root
of unity.
In terms of algebras the theorem says that if A is a projective Schur
algebra over k, then expŽ A mk k Ž ␮ .. divides the number of roots of unity
in k. We say that projective Schur algebras satisfy the exponent reduction
property.
To show how Theorem 1.2 follows from Theorem 1.3, take k s ⺡Ž x, t .,
the rational function field in two indeterminates over the rationals. Let
Kr⺡Ž t . be a regular extension Žover ⺡. which is cyclic of degree four Žsee,
for example, w7, p. 224x and let D s Ž K Ž x .rQŽ x, t ., ␴ , x . be the corresponding cyclic algebra.. It is clear that the order of D is not reduced by
any cyclotomic extension, that is, expŽ D mk k Ž ␮ .. s 4, whereas k does not
contain the fourth roots of unity. By Theorem 1.3, w D x f PSŽ k .. On the
other hand, D is split by a Kummer ⺪r2⺪ = ⺪r2⺪-extension, so w D x g
Br Ž Lrk ..
In w4x it is shown that certain projective Schur algebras, so-called radical
abelian algebras, satisfy the exponent reduction property. A projective
Schur algebra is radical if it is isomorphic to a crossed product algebra
A s Ž Krk, G, ␤ . where Krk is a Ž G-Galois. radical extension Ž K s
nr .
k Ž x 11r n1 , x 21r n 2 , . . . , x 1r
, x i g kU . and the class ␤ is represented by a
r
2-cocycle f whose values in K U are finite modulo kU . If G is abelian we
call A radical abelian. A radical algebra should be viewed as the ‘‘natural’’
way to construct projective Schur algebras. This is analogous to the
construction of cyclotomic algebras Žsee w10x..
It is conjectured in w1x that every element in PSŽ k . may be represented
by a radical algebra, even Žas conjectured in w4x. by a radical abelian
algebra. In this paper we show the stronger conjecture holds if the field k
has positive characteristic. In fact we show that every element in PSŽ k .
may be represented by the tensor product of algebras which are ‘‘natural’’
examples of radical abelian algebras.
Let A s Ž Krk, Gal Ž Krk ., ␣ . be a crossed product algebra over the
field k. It is well known that if Krk is cyclic, one may represent the class
␣ g H 2 Ž Gal Ž Krk ., K U . by a 2-cocycle whose values are in kU Žrather than
K U .. If in addition the field K can be embedded in a radical extension L
of k, then we may represent the algebra A by the crossed product algebra
Ž Lrk, Gal Ž Lrk ., infŽ ␣ .. where infŽ ␣ . denotes the image of ␣ under the
inflation map. The point of this is that also infŽ ␣ . may be represented by a
2-cocycle with values in kU Žin fact, the same set of values., so we see that
Ž Lrk, Gal Ž Lrk ., infŽ ␣ .. is a radical algebra. Note that if the extension
EXPONENT REDUCTION
359
Lrk is abelian, then Ž Lrk, Gal Ž Lrk ., infŽ ␣ .. is a radical abelian algebra.
We see that if an element w A x in Br Ž k . is split by a cyclic extension of k
which is contained in a radical Žresp. radical abelian . extension of k then
A is equivalent to a radical Žresp. radical abelian . algebra. Here we
consider two types of such algebras.
Type I. Symbol algebras. These are split by a cyclic Kummer extension of k.
Type II. Algebras that are split by a cyclic extension of k which is
contained in a cyclotomic extension of k. Note that algebras of type I and
type II are radical abelian.
THEOREM 1.4. Let k be a field of positi¨ e characteristic. Then e¨ ery
element in PSŽ k . may be represented by an algebra which is the tensor product
of algebras of types I and II.
Using the fact that a tensor product of radical Žradical abelian. algebras
is equivalent to a radical Žradical abelian . algebra Žsee w3, Lemma 2.4x,
where it is proved for radical algebras, but the same proof applies for
radical abelian algebras., we obtain:
COROLLARY 1.5. If char Ž k . / 0, e¨ ery element in PSŽ k . is represented by
a radical abelian algebra.
2. PROOFS OF THE RESULTS
We are given a projective Schur algebra A over k. By definition, AU
Žthe group of units of A. contains a subgroup ⌫ which spans A as a
k-vector space and which is finite modulo kU . We write A s k Ž ⌫ .. For any
subgroup H of ⌫ we may consider the subalgebra spanned by H and
denote it by k Ž H .. Recall that ⌫ is center by finite, so by a theorem due to
Schur, ⌫X , the commutator subgroup of ⌫, is finite. The subalgebras k Ž H .
may not be simple, but if we restrict ourselves to subgroups H, with
⌫X : H : ⌫ we have the following reduction w1x:
THEOREM 2.1. Let k Ž ⌫ . be a projecti¨ e Schur algebra o¨ er k. Then it is
Brauer equi¨ alent to a projecti¨ e Schur algebra k ŽU ., UrkU finite, and such
that for e¨ ery subgroup H of U with U X : H : U, k Ž H . is a simple algebra.
We say the algebra k ŽU . is reduced. Thus, for the proof of Theorem 1.3,
we shall assume that k Ž ⌫ . is a reduced algebra. Consider the subalgebra
k Ž ⌫X . and let K = k be its center. Since the group ⌫X is finite, it follows
that k Ž ⌫X . is a Schur algebra over K. Furthermore it is a simple component of the group algebra k⌫X and therefore K : k Ž ␨ . where k Ž ␨ . is a
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ALJADEFF AND SONN
cyclotomic extension of k Ž ␨ a root of unity. w2x. In particular, the family
⍀ [ k Ž H . : ⌫X : H : ⌫, k Ž H . is a Schur algebra over its center L H , and
L H is contained in a cyclotomic extension of k 4 is not empty.
Let k Ž H . be a maximal element in ⍀. Observe that such an element
exists because k Ž ⌫ . is finite dimensional over k. Let k Ž ␨ . be a cyclotomic
extension of k which contains L H s ZŽ k Ž H ... By Brauer’s Splitting Theorem w5, pp. 385, 418x, k Ž H . is split by a cyclotomic extension of L H , so we
may assume that k Ž ␨ . splits k Ž H .; that is, k Ž ␨ . mL H k Ž H . ( Mr Ž k Ž ␨ ..
for some integer r. We are to show that expŽ k Ž ⌫ . mk k Ž ␨ .. divides the
number of roots of unity in k.
To start with, note that the group ⌫ acts on k Ž H . by conjugation, hence
on its center L H .
LEMMA 2.2. The fixed field L⌫H of L H under ⌫ is k.
Proof. This is clear, since L⌫H ; ZŽ k Ž ⌫ .. s k.
Let Hˆ s C ⌫ Ž L H ., the centralizer of L H in ⌫. By the definition of Hˆ it
follows that ⌫X : H : Hˆ and L H : ZŽ k Ž Hˆ .. s L Hˆ . We claim that L H s
Hˆ
L Ĥ . Indeed, the group ⌫rHˆ acts faithfully on L H , and since L⌫r
s k we
H
see that < ⌫rHˆ < s w L H : k x. But ⌫rHˆ acts faithfully on L Hˆ as well, so by a
Hˆ
similar argument, we have L⌫r
s k. This implies w L Ĥ : k x s < ⌫rHˆ < s
Ĥ
w L H : k x, proving the claim.
In order to simplify notation let L s L H s L Hˆ .
PROPOSITION 2.3.
k Ž ⌫ . mk L is Brauer equi¨ alent to k Ž Hˆ ..
Proof. Consider k Ž ⌫ . as a module over k Ž Hˆ .. Let us show it is free of
rank < ⌫rHˆ <. Let g 1 , . . . , g n4 be representatives of the left cosets of Hˆ in
⌫. Clearly, they span k Ž ⌫ . as a left k Ž Hˆ . module. To show they are
independent we let w s z1 g i1 q z 2 g i 2 q ⭈⭈⭈ qz s g i s s 0 be a nontrivial relation of shortest length. Clearly s ) 1 since the g i are invertible. Take
y1 Ž
x g L with g i1 xgy1
such an element exists since ⌫rHˆ acts
i 1 / g i 2 xg i 2
faithfully on L., and consider
wx y g i1 Ž x . w s wx y g i1 xgy1
i1 w
s z1 g i1Ž x . g i1 q z 2 g i 2Ž x . g i 2 q ⭈⭈⭈ qz s g i sŽ x . g i s
y z1 g i1Ž x . g i1 q z 2 g i1Ž x . g i 2 q ⭈⭈⭈ qz s g i1Ž x . g i s
s z 2 g i 2Ž x . y g i1Ž x . g i 2 q z 3 g i 3Ž x . y g i1Ž x . g i 3 q ⭈⭈⭈
qz s g i sŽ x . y g i1Ž x . g i s s 0,
yielding a shorter nontrivial relation, since 0 / g i 2Ž x . y g i1Ž x . g L, a contradiction.
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EXPONENT REDUCTION
To prove the proposition we show
L mk k Ž ⌫ . ( End kŽ Hˆ . Ž k Ž ⌫ . . s Hom kŽ Hˆ . Ž k Ž ⌫ . , k Ž ⌫ . . ( Mn Ž k Ž Hˆ . .
Ž k Ž ⌫ . with the left k Ž Hˆ . structure .. Indeed, the algebra k Ž ⌫ . acts on k Ž ⌫ .
from the right and L acts on k Ž ⌫ . from the left. Clearly the actions
commute and both commute with the k Ž Hˆ . left action. ŽOf course, here
we are using that L s ZŽ k Ž Hˆ ... This gives a nontrivial homomorphism ␩
from L mk k Ž ⌫ . into End kŽ Hˆ .Ž k Ž ⌫ ... The map ␩ is 1᎐1 Žsince L mk k Ž ⌫ . is
simple. and therefore surjective Žthe dimensions over k are equal.. This
completes the proof of the proposition.
Observe that the algebras k Ž H . and k Ž Hˆ . have the same center L, and
so Žby the double centralizer theorem. we have k Ž Hˆ . , k Ž H . mL A Ž A
central simple over L.. Theorem 1.3 will follow if we show that expŽ A. Žin
Br Ž L.. divides the number of roots of unity in k. We start with the
following key lemma.
LEMMA 2.4. Let z g Hˆ and let n be its order modulo k Ž H .U . Then n
di¨ ides the number of roots of unity in k.
Proof. We assume the lemma is false and get a contradiction. Taking a
power of z we may, for some prime p, assume the order of z modulo
k Ž H .U is p rq1 and k contains the p r th roots of unity but not the p rq1 th
root of unity, r G 0.
Consider the action of z on k Ž H . induced by conjugation. Since this
action centralizes L, by the Skolem᎐Noether Theorem there is an element
a g k Ž H .U such that w s zay1 centralizes k Ž H .. In particular, it centralizes a, so z and a commute in k Ž Hˆ .. It follows that the element w is in
the center of the subalgebra B s k Ž² H, z :. s k Ž w .Ž H . s LŽ w .Ž H . Žin
k Ž Hˆ ... Clearly LŽ w . is contained in the center of B. In fact LŽ w . is
precisely the center of B since there is an obvious map of algebras from
LŽ w . mL k Ž H . Žsimple with center LŽ w .. onto B.
rq 1
Claim. The element w satisfies an equation of the form X p y e s 0
where e g LU . Furthermore p rq1 is the smallest possible. To see this,
recall that w centralizes a so that a, z, w commute. We therefore have
rq 1
rq 1
rq 1
rq 1
rq 1
w p s Ž zay1 . p s z p ayp g k Ž H .U . But w p
centralizes k Ž H .,
rq 1
hence w p g L s ZŽ k Ž H ... Finally, it is easy to see that the order of w
modulo LU equals the order of z modulo k Ž H .U Žs p rq1 .. This proves the
claim. Observe that the algebra B is a Schur algebra over its center LŽ w .
since k Ž H . is, and moreover it is of the form k Ž H˜ ., where H˜ s ² H,
i
i
z :. Set B0 s B and for i s 1, . . . , r let Bi s k Ž² H, z p :. s k Ž w p .Ž H . s
i
i
LŽ w p .Ž H .. As for B, one shows easily that LŽ w p . s ZŽ Bi .. Also, note
362
ALJADEFF AND SONN
that Br strictly contains k Ž H ., so we will reach a contradiction to the
r
maximality of k Ž H . if we show that LŽ w p . is contained in a cyclotomic
extension of k. Let us show that this is indeed the case.
First recall that the field LŽ w . is the center of the algebra B s
k Ž² H, z :.. By w2, Corollary 2.3x, LŽ w . is contained in the composite of the
maximal cyclotomic extension k Ž ␮ . of k and a Žfinite. Kummer extension
k ŽU . of k. Since k contains no primitive p rq1 root of unity, the p-primary
component of the Galois group Gal Ž k ŽU .rk . has exponent F p r.
It follows that the p-primary component of Gal Ž k Ž ␮ , U .rk Ž ␮ .. has
exponent at most p r. On the other hand the field k Ž ␮ , w . is a Kummer
r
extension of k Ž ␮ . and is contained in k Ž ␮ , U .. Hence if w p f k Ž ␮ .,
rq1
Gal Ž k Ž ␮ , w .rk Ž ␮ .. is cyclic of order p . This is impossible of course. It
r
r
follows that w p g k Ž ␮ . and hence LŽ w p . : k Ž ␮ .. This completes the
proof of the lemma.
Our next step will be to decompose the algebra k Ž Hˆ . as a tensor
product k Ž H . mL A, with expŽ A. dividing the number of roots of unity in
k.
To do this recall that PAbŽ F . is the subgroup of PSŽ F . consisting of
classes which may be represented by a projective Schur algebra F Ž ⌫ . and
⌫rF U s G abelian. In this case we say that F Ž ⌫ . is of abelian type.
Obviously, the natural examples are the symbol algebras. We recall from
w3x the following.
PROPOSITION 2.5. If F Ž ⌫ . is a projecti¨ e Schur algebra of abelian type,
then it is Brauer equi¨ alent to the tensor product of symbol algebras. Furthermore, if expŽ G . s n, then the exponent of F Ž ⌫ . in Br Ž F . di¨ ides n.
LEMMA 2.6. Let k Ž H . and k Ž Hˆ . be as abo¨ e. Then k Ž Hˆ . ( k Ž H . mL
LŽ ⌳ . where LŽ ⌳ . is a projecti¨ e Schur algebra of abelian type and expŽ ⌳rLU .
di¨ ides the number of roots of unity in k. Furthermore, LŽ ⌳ . is isomorphic to
a product of symbol algebras.
Let us postpone the proof of the lemma and complete the proof of
Theorem 1.3.
By Lemma 2.6, expŽ LŽ ⌳ .. Žas an element in Br Ž L.. divides expŽ ⌳rLU .,
hence expŽ LŽ ⌳ .. divides the number of roots of unity in k. But k Ž ⌫ . mk
k Ž ␮ . ; k Ž Hˆ . mL k Ž ␮ . Ž L : k Ž ␮ . and Proposition 2.3; ; denotes Brauer
equivalence. ( k Ž H . mL LŽ ⌳ . mL k Ž ␮ . ; LŽ ⌳ . mL k Ž ␮ . Ž k Ž H . is Schur
over L, hence is split by k Ž ␮ ...
This implies that expŽ k Ž ⌫ . mk k Ž ␮ .. s expŽ LŽ ⌳ . mL k Ž ␮ .. which divides expŽ LŽ ⌳ ... It follows that k Ž ⌫ . mk k Ž ␮ . divides the number of roots
of unity in k and the theorem is proved.
Proof of Lemma 2.6 Ž See w3, Lemma 2.3x... Recall that Hˆ normalizes
ˆ Ž H .U of the units of
k Ž H .U ; hence we may consider the subgroup Hk
EXPONENT REDUCTION
363
ˆ Ž H .U . Since ⌳ l k Ž H .U s
k Ž Hˆ .. Let ⌳ be the centralizer of k Ž H . in Hk
ˆ Ž H .U rk Ž H .U which is a quotient of HrL
ˆ U
LU , it follows that ⌳rLU ; Hk
U
ˆ
l H. This implies ⌳rL is finite. Furthermore, the commutator subgroup
HˆX of Hˆ is contained in k Ž H .U Žsince HˆX ; ⌫X ; H ., so ⌳rLU is abelian.
Finally, we recall from Lemma 2.4 that the order of any element in Hˆ
modulo k Ž H .U , hence expŽ ⌳rLU ., divides the number of roots of unity in
k. By Proposition 2.5 it follows that the order of LŽ ⌳ . in Br Ž L. divides
expŽ ⌳rLU ., hence it also divides the number of roots of unity in k. In the
proof of Proposition 2.5, it is shown that LŽ ⌳ . is isomorphic to a product
of symbol algebras. To complete the proof of Lemma 2.6, we need to show
that k Ž Hˆ . s k Ž H . mL LŽ ⌳ .. By the double centralizer theorem, we have
k Ž Hˆ . = k Ž H . mL LŽ ⌳ . s k Ž H . LŽ ⌳ .. In order to prove the reverse incluˆ z normalizes k Ž H . and centralizes its center
sion, take an element z g H.
L. Therefore, there is an element eŽ z . g k Ž H .U such that zy1 eŽ z . centralizes k Ž H .. This puts zy1 eŽ z . in ⌳, and the lemma is proved.
COROLLARY 2.7. If the field k contains all roots of unity then e¨ ery
element in PSŽ k . is equi¨ alent to a product of symbol algebras.
ŽThis is of course a direct consequence of the Merkurev᎐Suslin Theorem..
Proof. Let k Ž ⌫ . be a projective Schur algebra. We may assume it is
reduced. If k contains all roots of unity then the field L above must be k,
so the algebra k Ž Hˆ . coincides with the entire algebra k Ž ⌫ .. From the
proof above, it follows that k Ž ⌫ . is the tensor product of a Schur algebra
k Ž H . over k and symbol algebras. But k Ž H . must be split ŽBrauer’s
theorem., hence k Ž ⌫ . ; product of symbol algebras.
We turn now to the proof of Theorem 1.4.
Let k be a field of characteristic p ) 0 and let A s k Ž ⌫ . be a projective
Schur algebra over k. We are to show that A is represented by the tensor
product of symbol algebras and an algebra which is split by a cyclotomic
extension of k Žnote that in positive characteristic every finite cyclotomic
extension is cyclic.. Since the tensor product of such algebras Žtensor
product of symbol algebras and an algebra which is split by a cyclotomic
extension of k . is again such an algebra, we may assume expŽ A. s q t ,
where q is a prime number and t ) 0. If k contains no nontrivial qth
roots of unity Žin particular if q s p ., A is split by a cyclotomic extension
by Theorem 1.1 Ž k has no Kummer q-extension.. On the other hand if k
contains all q-power roots of unity, then A is similar to a tensor product of
symbol algebras by w8x or Corollary 2.7. We therefore assume k contains a
primitive q r th root of unity Ž r ) 0. but not a primitive q rq1 th root of
unity. By exponent reduction ŽTheorem 1.3. there is a finite cyclotomic
extension k Ž ␨ . of k such that expŽ A mk k Ž ␨ .. s q s, 0 F s F r. It follows
364
ALJADEFF AND SONN
that Amq is split by k Ž ␨ . and is therefore equivalent to a cyclic algebra
Ž k Ž ␨ .rk, ␴ , a. where Žby abuse of notation. the 2-cocycle is determined by
the relation ␴ n s a, n s w k Ž ␨ . : k x. The key observation here is that the
s
algebra Amq has a q s th root in Br Ž k . which is split by a cyclotomic
extension of k. In other words, there is a cyclic algebra B s Ž k Ž ␨ 1 .rk, ␶ , b .
s
s
with Bmq ; Amq . Of course B is a radical abelian algebra. Let us
complete the proof of the theorem assuming such a B exists. Consider the
s
algebra C s By1 mk A. Clearly w C x q s 1 in Br Ž k .. Furthermore k cons
tains a primitive q th root of unity and so by the Merkurev᎐Suslin
Theorem, C may be represented by a product of symbol algebras. The
theorem is now proved since A ; B mk C. It remains to show the existence
of B. Keeping in mind that charŽ k . s p, we see that the cyclic cyclotomic
extension k Ž ␨ .rk can be embedded into a cyclic cyclotomic extension
k Ž ␨ 1 .rk of degree q s n. By w9, p. 262x, the algebra B s Ž k Ž ␨ 1 .rk, ␶ , a.
Žsame a as in A. has the desired property.
s
Remark. From the proof we see that if charŽ k . / 0, then a central
simple k-algebra has the exponent reduction property if and only if it is a
projective Schur algebra over k.
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