Centralp-Extensions of (p, p,…,p)-Type Galois Groups

JOURNAL OF ALGEBRA ARTICLE NO.

186, 277]298 Ž1996.

0373

Central p-Extensions of Ž p, p, . . . , p . -Type Galois Groups John R. SwallowU Department of Mathematics, Da¨ idson College, Da¨ idson, North Carolina 28036 Communicated by Mel¨ in Hochster Received August 2, 1994

Let p be a prime number, K a field with characteristic not p and containing the pth roots of unity, and ErK an abelian exponent p Galois extension. We prove explicit formulas for the construction of fields NrK with Galois group a central p-extension of GalŽ ErK .. These formulas do not require the solution of a linear system of equations in the field extension, as do the formulas of Massy w J. Algebra 109 Ž1987., 508]535x. In our study we develop generalizations of theorems of Serre and Frohlich on representing obstructions to embedding problems and theorems of ¨ Crespo on explicitly constructing solution fields. Q 1996 Academic Press, Inc.

1. INTRODUCTION Let ErK be a Galois extension with group G and suppose that we have an exact sequence

˜ª G ª 1 1 ª ZrpZ ª G ˜ as a central ZrpZ-extension of G. A classical question which describes G is to ask whether E can be embedded in a field EX such that EXrK is ˜ If so, one asks how to construct such an extension EX Galois with group G. from E. Particular embedding problems, such as that of constructing a quaternion-group extension from a Klein 4-group extension, were considered by Dedekind wDex and Witt wWix. Various results for small p-group embedding problems appeared in the 1970s, owing to Damey and Payan U

A portion ŽSection 2. of this work is derived from the author’s thesis, written under the thoughtful supervision of Walter Feit. Research supported in part by NSF grant DMS-9108148. 277 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

278

JOHN R. SWALLOW

wDaPx, Damey and Martinet wDaMrx, and Massy and Nguyen-Quang-Do wMaNx. More recently, several new methods have been developed for the solution of embedding problems for larger groups. A recent survey on the state of solving central embedding problems via primarily algebraic methods may be found in wGSSx. Serre wSe1x proved that a necessary and sufficient condition for the positive solution of certain embedding problems with p s 2 is the triviality of the Hasse]Witt invariant of a particular trace form associated to a subfield of ErK. Serre also asked how one would construct EX if such an embedding problem is solvable. A theorem of Crespo wCr1x answers this question with a method for the explicit construction of EX , given an isomorphism from the trace form to one of a list of quadratic forms having the same rank, discriminant, and Hasse]Witt invariant as a sum of squares quadratic form. Over global fields, the Hasse]Witt invariant needs to be checked at only a finite number of completions, and the list of quadratic forms necessary for Crespo’s theorem is finite, depending only on signatures. Although Serre’s theorem holds only for certain G, Frohlich’s general¨ ization wFrx widens the applicability of the concept, showing that for more embedding problems Žstill with p s 2. the necessary and sufficient condition can be viewed as the triviality of a product of Hesse]Witt invariants of certain quadratic forms associated to subfields of ErK. Crespo’s method can be shown to hold for Frohlich’s generalization wCr2x. Frohlich’s ¨ ¨ paper indicates analogous results for p ) 2, using central simple algebras in place of quadratic forms and the reduced norm in place of the spinor norm, and Crespo again extends her method for this analogue wCr3x. For these results, one requires that the ground field K be of characteristic not p and contain the pth roots of unity. These generalizations of Serre’s theorem, however, have not extended so far as to apply to all central p-extensions. It is unclear whether all embedding problems with p s 2 are covered by Frohlich’s generalization ¨ unless the ground field contains enough roots of unity; one must find homomorphisms from G to an orthogonal group of a quadratic K-form. The restrictions on the p ) 2 analogue are more severe: one must consider homomorphisms from G into elements of reduced norm 1 of a central simple K-algebra of degree p, and frequently there are few such maps. Also, even when one can find such a homomorphism corresponding to the p-extension of G, determining the central simple algebra which results under Galois cohomology is a nontrivial problem. In this paper we address some of these concerns. In Section 2 we present a generalization of Frohlich’s method of representing obstructions ¨ to central p-embedding problems, in which we extend the method naturally to accommodate 1-cocycles in place of the homomorphisms from the

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group G to the group of elements of reduced norm 1 of a central simple algebra of degree p. We then expand Crespo’s explicit construction theorems accordingly. These results widen still further the applicability of the Serre method. Our method still requires the explicit determination of a central simple algebra corresponding under Galois cohomology to an element of H 1 Ž G, PSL nŽ E .., and this step remains difficult in general. As an application of our results, we treat in Section 4 the case of p-extensions of abelian exponent p groups, considered by Massy wMa1, Ma2x. The Galois cohomology step is circumvented by necessary and sufficient conditions of wMa2x for the existence of solutions to embedding problems of this sort. Using Massy’s normic reciprocity law and numerical decomposition of elements of H 2 ŽŽZrpZ. n, Fp ., we derive formulas based on the required field elements. We recall what we need of Massy’s results in Section 3, and in Section 6 we provide some examples. Our method improves upon the explicit constructions for n ) 2 in wMa1, Ma2x in that our formulas do not depend on the solution of a nontrivial system of linear equations in the field extension, viewed as a vector space over the ground field. Our method does not appeal to Hilbert’s Satz 90; rather, we use the nondegeneracy of the trace form to establish a finite list Žof length at most p n . of purported solutions to the embedding problem, such that at least one element of the list is nonzero and such that any nonzero element of the list is in fact a solution. For small embedding problems this list can be shortened, and doing so is the aim of Section 5. The method presented here is one of several developed in the author’s thesis wSw1x. The expansion of Frohlich’s quadratic form method Žthe ¨ w x. ‘‘orthogonal case’’ in Sw1 to include 1-cocycles was introduced in wSw2x, and the present paper outlines the ‘‘reduced projective case’’ from wSw1x. The ‘‘full projective case’’ from wSw1x, in which we drop the requirement that the elements in a central simple algebra lying in the image of the 1-cocycle have reduced norm 1 and we replace the reduced norm with an appropriate set of polynomials on the matrix coefficients of the central simple algebra, is presented in wSw3x.

2. TWISTED PROJECTIVE REPRESENTATIONS OF GALOIS GROUPS AND EXPLICIT CONSTRUCTION In what follows p will denote a prime number, K a field of characteristic not p which contains the pth roots of unity m p , and K s the separable closure of K. We fix throughout a primitive pth root of unity j p .

˜ ª G ª 1 be an extension of finite THEOREM 1. Let 1 ª ZrpZ ª G groups, with G s GalŽ ErK . for fields ErK. Let A be a finite-dimensional

280

JOHN R. SWALLOW

central simple algebra o¨ er K of degree p and t: G ª PSLŽ A m E . a 1-cocycle such that the image of t under the coboundary homomorphism H 1 Ž G, PSLŽ A m E .. ª H 2 Ž G, Fp . associated to the sequence at the algebraic closure 1 ª Fp ª SL Ž A m K s . ª PSL Ž A m K s . ª 1

˜ ª G. Ž Here SLŽ A. is the class c g H 2 Ž G, Fp . of the embedding problem G denotes the elements of reduced norm 1 of A, and PSLŽ A. denotes the projecti¨ e group of these elements.. Let A˜ be the algebra gi¨ en by Galois cohomology associated to A and t. Then the homomorphism GalŽ K srK . ª ˜ if and only if the quotient of the GalŽ ErK . can be lifted to factor through G class of A and the class of A˜ in the p-torsion of the Brauer group H 2 ŽGalŽ K srK ., Fp . ( Br p Ž K . is tri¨ ial. Proof. We follow wFrx, Appendix. The condition on lifting the homomorphism of Galois theory GalŽ K srK . ª GalŽ ErK . is equivalent to the triviality of the image of the class of the group extension in H 2 Ž G, Fp . when inflated to H 2 ŽGalŽ K srK ., Fp . wHox. Since the class c g H 2 Ž G, Fp . of the group extension is the image of the class of A˜ in the coboundary homomorphism given above, the image of the quotient of the classes of A and A˜ under the coboundary map H 1 Ž G, PSL p Ž E .. ª H 2 Ž G, Fp . must yield c. The coboundary map is an isomorphism at the algebraic closure wSe2x; hence the if and only if condition. We define p-cyclic algebra generators to mean two elements i, j of a central simple algebra of dimension p 2 over K such that i p s a g K U , j p s b g K U , and ji s j p ij. THEOREM 2. Assume the conditions of Theorem 1 and assume that the quotient of the classes of A and A˜ is tri¨ ial. Then there exist an isomorphism of K-algebras h: A˜ ª A, an E-automorphism f : VA m E ª VA m E of the underlying ¨ ector space VA of A such that both f Ž A. ( A˜ and f g fy1 s t Ž g . for g g G, and an in¨ ertible element z g A m E such that f Ž a. z s zhŽ a. for a g A. If c / 0 g H 2 Ž G, Fp ., then the complete set of solution fields EX is  EŽŽ k NrdŽ z ..1r p .: k g K U 4 , where NrdŽ z . is the reduced norm of z. If c s 0 in H 2 Ž G, Fp ., then the complete set of solution fields EX is  EŽ r 1r p .: r g K U , r 1r p f E4 . If A is a p-cyclic algebra with generators i, j, and f is altered by a K-automorphism so that i and j are sent to the corresponding ˜ then one of a finite number of p-cyclic algebra generators of f Ž A. ( A, alterations of h will lea¨ e h an isomorphism and insure that the element zs

Ý

f Ž i.

n1

f Ž j.

n2

h Ž j.

n 1 , n 2g 0, 1, . . . , py1 4

is in¨ ertible and satisfies f Ž a. z s zhŽ a. for a g A.

yn 2

h Ž i.

yn 1

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Proof. We follow wCr1, Cr3x, with generalizations from wSw1x. If the quotient of classes of A and A˜ is trivial, A (K A˜ since they have the same degree. Therefore there exists a K-isomorphism h. Let tX be the element of H 1 Ž G, GLŽ VA m E .. corresponding to t under the conjugation-action inclusion PGLŽ A m E . ª GLŽ VA m E ., where VA is the underlying vector space of the algebra A over K. Since H 1 Ž G, GL Ž VA m E . . s 1 wSe2x, there exists an E-isomorphism of vector spaces f : VA m E ª VA m E ˜ The images of A under f such that f g fy1 s t Ž g ., g g G, and f Ž A. ( A. and h inside A are isomorphic; hence by Noether]Skolem there exists an invertible z such that zy1 f Ž a. z s hŽ a.. Let u be a section of PSLŽ A m E . in SLŽ A m E . and let x g s uŽ t Ž g .., g g G. Then we have for any basis  e i j 4 of A, xy1 g ei j x g s t Ž g . Ž ei j . ,

g g G,

and X

X Ž g , g X . ¬ x g X x gg xy1 gg ,

g , gX g G

is the 2-cocycle corresponding to c. If c s 0, then we have the complete set of solution fields as given in the theorem. Otherwise, we show that NrdŽ z . satisfies the well-known condition on an element of the field to have pth root generate an EXrE when c / 0. Let  e iX , e jX 4 be p-cyclic algebra generators of the K-algebra A˜ generated by  f Ži., f Žj.4 , and let  e i , e j 4 be such that e iX s f Ž e i ., e jX s f Ž e j .. We have that g

xy1 g f Ž e . xg s f Ž e . ,

g g G, e g  e i , e j 4 .

Set bg s x g z g zy1 , g g G. We claim that the bg are in EU and satisfy X Ž . Ž . g s hŽ e i ., hŽ e j . g s hŽ e j . for g g G we X bg X by1 g g s c g, g . Since h e i have that bgg

X

g

g

g g x g z g h Ž e . s x g z g h Ž e . s x g f Ž e . xy1 g xg z s f Ž e . xg z ,

g g G, e g  e i , e j 4 .

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JOHN R. SWALLOW

Hence x g z g zy1 commutes with f Ž e i . and f Ž e j .: x g z g zy1 f Ž e . s x g z g h Ž e . zy1 s f Ž e . x g z g zy1 ,

g g G, e g  e i , e j 4 .

Since f Ž e i . and f Ž e j . generate A as an E-algebra, the bg are in the center and hence in E. Since z and the x g are nonzero, the bg are also and hence invertible. Now we simply calculate X

X

X

g g g yg X bgg bg X by1 z gg s Ž xg z X

X

s Ž x gg z g g zyg

X

X

X

X

. Ž xg

X

X z g zy1 .Ž zzyg g xy1 gg .

. Ž xg

X

g g g yg X z g zyg g xy1 z gg . s Ž xg z

X

X

X

X

X

X

X g y1X X X X s Ž x gg xy1 g g xg . s Ž xg xg xg g . s cŽ g , g . ,

X

X

X

. Ž z g zyg g xy1 gg xg . X

X

g , g X g G,

where parentheses surround an element of E which implies that the multiplication may be rotated Ži.e., a ? b ? c s c ? a ? b .. Now we have the necessary relation on the pth root of an element of E to generate EXrE. . s 1, that The relation z g zy1 s bg xy1 implies, given NrdŽ x g . s NrdŽ xy1 g g g

Nrd Ž z . s bgp Nrd Ž z . , giving us that NrdŽ z . is such an element. Finally we show that z can be computed if A has the presentation given in the theorem. Assume this presentation. Let h be a K-isomorphism of algebras from A˜ to A. If the element z defined in the theorem is 0, consider the resulting h which one gets by sending i to the product of its old h-image and a power of j p . If the new z’s are also zero for every power of j p , we will have

Ý

f Ž j.

n2

h Ž j.

n2

s 0.

n 2g 0, 1, . . . , py1 .

We then change h so that j is sent to the product of its old h-image and powers of j p . We cannot get zero for every alteration, lest we derive the fact that 1 s 0. Hence some alteration must have resulted in a nonzero z. We claim that f Ž e . z s zhŽ e ., e g  i, j4 . One need only consider pairs of summands in the definition of z such that the n j differ only at a fixed n i and that these n i differ by 1. If n 1 s k, for instance, then this summand multiplied on the left by f Ži. is identical to the other summand Žthat of n 1 s k q 1. with hŽi. multiplied on the right and vice versa. Since our condition is valid on i and j, and f and h are algebra homomorphisms, the condition is valid for all a g A.

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3. EMBEDDING PROBLEMS, 2-COCYCLES, AND NORMIC RECIPROCITY In what follows ErK is a Galois extension with group G s ŽZrpZ. n and the pth root notation ?1r p denotes an arbitrary but fixed choice. The central extensions of GalŽ ErK . by ZrpZ are classified by the cohomology group H 2 Ž G, ZrpZ. ( H 2 Ž G, Fp ., where the action on ZrpZ is trivial. We first recall Massy’s numerical decomposition of this group: THEOREM 3 wMa2x. Let c g H 2 Ž G, Fp .. Define c# to be the bilinear alternating form on G gi¨ en by c#Ž g 1 , g 2 . s cŽ g 1 , g 2 . y cŽ g 2 , g 1 ., g 1 , g 2 g G. Define cU by the rule cU Ž g . s Ý r mod p cŽ g r , g ., g g G. The function cU is a quadratic form on G for p s 2 and a linear form on G for p / 2. Both cU and c# are independent of the choice of c g c. Let GX s GrKerŽ cU < rad c# ., dX s dim F p GX , and 2 r be the rank of c#. We ha¨ e dX s 2 r q v , with v g  0, 14 . For any elements a, b g K U l EU prK U p , define Ž a, b . g H 2 Ž G, Fp . by the usual cup product of the linear forms Ž a., Ž b . g H 1 Ž G, Fp . defined by the rule g Ž a1r p .ra1r p s j pŽ a.Ž g .. Define also ŽŽ a.. g H 2 Ž G, Fp . to be j p ? d ŽŽ1rp .Ž a.Z ., where d is the coboundary H 1 Ž G, QrZ. ª H 2 Ž G, Z., Ž a.Z is a lifting of Ž a. to H 1 Ž G, Z., and the ? represents the cup-product multiplication Hˆ 0 Ž G, m p . = H 2 Ž G, Z. ª H 2 Ž g, Fp .. Ži. If dX s 0, then c s 0. Žii. If v s 1, then there exist elements a i , 0 F i F r, bi if r / 0, 1 F i F r, which are linearly independent in K U l EU prK U p and such that r

c s Ž Ž a0 . . q

Ý Ž ai , bi . . is1

Žiii. If v s 0, then there exist n g  0, 14 and elements a i , bi , 1 F i F r, which are linearly independent in K U l EU prK U p and such that

¡n Ž Ž a b . . q Ý Ž a , b . , r

c s~

i

if p s 2,

Ý Ž ai , bi . ,

if p / 2.

i

1 1

is1

¢

n Ž Ž a1 . . q

r is1

˜ª G Theorem 3 gives the obstruction to the embedding problem G given by the 2-cohomology class c. By Theorem 2, if c / 0, there is a positive solution to the embedding problem if and only if the inflation of c to GalŽ K srK . is trivial. The inflation of ŽŽ a.. is the p-cyclic algebra Ž a, j p . and that of Ž a, b . is the p-cyclic algebra Ž a, b . p . Hence the condition is that

284

JOHN R. SWALLOW

the tensor product of various p-cyclic algebras is trivial in the Brauer group of K. There is an equivalent condition for our extensions, however, which Massy calls normic reciprocity: THEOREM 4 wMa2x. Theorem 3.

Let the numerical decomposition of c / 0 be as in

r Case I. n s 0 or v s 1. Let  si 4is1y v be the K-automorphisms of 1r p r 1r p p L [ K Ž a i 4is1y v . gi¨ en by si Ž a j . s j pd i j a1r . Then the embedding probj lem associated with c has a positi¨ e solution if and only if there exist elements x i g LU , 1 y v F i F r, such that

py 1

x 01q s 0qs 0 q ??? q s 0 2

s iqs i q ??? q s i x 1q i 2

s jp ,

py 1

if v s 1;

s bi ,

x js iy1

s

1 F i F r;

x is jy1 ,

1 y v F i, j F r.

Case II. v s 0, n s 1, p / 2. Let  si 41r be the K-automorphisms of p4 r . p Ž 1r p . s j pd i j a1r L [ K Ž a1r . Then the embedding problem i 1 gi¨ en by si a j j associated with c has a positi¨ e solution if and only if there exist elements x i g LU , 1 F i F r, such that py 1

x 11q s 1qs 1 q ??? q s 1 2

s iqs i q ??? q s i x 1q i 2

py 1

s j p b1 ; s bi ,

2 F i F r;

x js iy1 s x is jy1 ,

1 F i, j F r.

Case III. v s 0, n s 1, p s 2. Let  si 41 F i F r , t 1 be the K-automor41 F i F r j  b11r2 4. gi¨ en by si Ž a1r2 . s Žy1. d i j a1r2 phisms of L [ K Ž a1r2 i j j , 1r2 . 1r2 1r2 . 1r2 1r2 . Ž Ž Ž si b 1 s b 1 , t 1 a i s a i , 1 F i, j F r, and t 1 b1 s yb1r2 1 . Then the embedding problem associated with c has a positi¨ e solution if and only if there exist elements x i g LU , 1 F i F r, and y g LU such that si x 1q s bi , i

2 F i F r;

x js iy1 s x is jy1 ,

1 F i F r, 2 F j F r;

x tj 1y1 s y s jy1 ,

2 F j F r;

y 1q t 1 s x 11q s 1 s y1,

y s1y1 s yx 1t 1y1 .

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OF GALOIS GROUPS

4. EXPLICIT CONSTRUCTION FOR & n Ž ZrpZ . ª ŽZrpZ. n s GalŽ ErK . In order to follow the method of Section 2, we take the split central simple algebra A [ Ž1, 1. p of dimension p 2 with generators and relations i p s 1, j p s 1, ji s j p ij, and we construct a 1-cocycle in Z 1 Ž G, PSLŽ A m E .. satisfying two properties: First, that under the coboundary H 1 Ž G, PSLŽ A m E .. ª H 2 Ž G, Fp . the class of our cocycle is sent to c. Second, that using Theorem 4 we can describe A˜ and a K-isomorphism from A˜ to A. THEOREM 5. Let c / 0 ha¨ e a numerical decomposition as in Theorem 3 and let the field L, the elements x i corresponding to the sol¨ ability of c, and the K-automorphisms si of L, extended to be in¨ ariant on each bj1r p , 1 F j F r, be as defined in Theorem 4. Let t i , 1 F i F r, be the K-automorr . phisms of LX [ LŽ bi1r p 4is1 gi¨ en by t i Ž bj1r p . s j pd i j bj1r p , 1 F i, j F r, and 1r p 1r p t i Ž a j . s a j , 1 F i F r, 1 y v F j F r. For Cases I and II of Theorem 4, define a 1-cocycle t: G ª PSLŽ A m E . as the inflation to G of the 1-cocycle tX g H X Ž²  si 4 is1y v j  t i 4 is1: , PSL Ž A m LX . . , r

r

p gi¨ en as follows, where 0 F m i , mXj F p y 1, and b1r [ 1 if v s 1: 0

tX

ž

r

r

Ł

si m i Ł t jm j

is1y v

X

r

s

/

js1

X

Ł

is1y v

j pm i m i

py2 py1

žÝ Ý

l jyk p

bi1r p px i

ks0 ls0 py1

q

Ý

py1.l jyŽ p

il

x ipy 1 bi1r p pbi

ls0

r ms iy1 q ŽÝ m qs 0 s i . Ł ss iq 1 s s

i

l

/

.

For Case III of Theorem 4, define a 1-cocycle t: G ª PSLŽ A m E . as the inflation to G of the 1-cocycle tX gi¨ en as follows, where 0 F m i , mXj F 1: tX s 1m 1

ž ž

r

Ł si m

i

is2

žŁŽ

y1 .

js2

X

mimi

is2

r

=Ł is2

=

ž

ž

X j

Ł t jm

/ ž

r

s

r

X

t 1m 1

/ž

//

x 1 y x 1s 1 q x 1 i q x 1s 1 i 2

x i q x is i q x i i y x is i i 2 bi1r2

y y y t 1 q yi q y t 1 i 2

X

/

r ms m 19 m i Ž Ł ss iq 1 s s . t 1

/

X m1

m 1t 1

/

X

r ms m1 m 1Ž Ł ss 2 s s .t 1

.

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JOHN R. SWALLOW

The image of the class of t in H 1 Ž G, PSLŽ A m E .. under the coboundary homomorphism H 1 Ž G, PSLŽ A m E .. ª H 2 Ž G, Fp . is c. Proof. First, we claim that the reduced norm of the image of t is identically 1. The reduced norm of any constant is its pth power, so constant multiples of the j p do not affect the norm. Also, the reduced norm of any expression of the form py1 py1

žÝ Ý

l jyk p XŽ k.

p

ks0 ls0

il

/

py 1 Ž . is the product Ł ks 0 X k . In our case, using the relations among the x i of Theorem 4, we find that the reduced norm of each such expression is 1. Now we show that t for Cases I and II satisfies the 1-cocycle condition. We begin with the left X side of the 1-cocycle condition tŽ ˜ g . t Ž g . g˜, where X r mi r mj r m ˜iŁr t m ˜ j: g s Ł is1y s Ł t and g s Ł s ˜ v i js1 j is1y v i js1 j

r

t

ž

Ł

r

is1y v

si

m ˜i

Ł

js1

r

s

r

X

t jm˜ j X

j pm i m i

Ł

is1y v

/ž t

r

si

Ł

is1y v

mi

žÝ Ý

l jyk p

ks0 ls0 py1

q

j pl

Ý

=

Ł

j pm˜ i m˜ i

pbi

q

Ý

r

ž

X

Ł

is1y v r

=

X

j pm i m iqm i m˜ i

Ł

is1y v

jp

Ý Ý py1

j pl

Ý ls0

r

=

Ł

is1y v

l jyk p

pbi

bi1r p

ks0 ls0

q

X

j pm˜ i m˜ i

il r m ˜ iy1 q ˜s ŽÝm qs 0 s i . Ł ss iq 1 s s

il

/

/

py2 py1

ž

/

x py 1 bi1r p l i

ls0

s

i

px i

ks0 ls0 py1

r ms r m m ˜s r ˜ j9 iy 1 q ŽÝm qs 0 s i . Ł ss iq 1 s s Ł ss 1 s s Ł js 1t j

l

bi1r p

l jyk p

žÝ Ý

/

il

px i

py2 py1

X

is1y v

bi1r p

x ipy1 bi1r p

ls0 r

Ł

js1

py2 py1

r m r m ˜i ˜ j9 Ł is 1y v s i Ł js 1t j

X

t jm j

px i

il

x ipy1 bi1r p pbi

py2 py1

žÝ Ý

ks0 ls0

l jyk p

qq m ms ˜ i.Ł r iy 1 ŽÝm Ł irXs 1y v s iXm˜ iX qs 0 s i ss iq 1 s s

i

l

/

bi1r p px i

il

p-EXTENSIONS

CENTRAL

py1

q

x ipy 1 bi1r p

j pl

Ý

pbi

ls0

287

OF GALOIS GROUPS r m ˜ iy1 q ˜s ŽÝm qs 0 s i . Ł ss iq 1 s s

i

/

l

.

Now consider the following relation, derived from the x is jy1 s x js iy1 relations of Theorem 4: py2 py1 l jyk p

bi1r p

py1

žÝ Ý žÝ Ý žÝ Ý žÝ Ý ks0 ls0

px i

py2 py1

=

ks0 ls0

i q l

Ý

pbi

ls0

py1

p X b1r yk l i jp il q px iX

py2 py1

l jyk p

s

x ipy1 bi1r p

j pl

ks0 ls0

py2 py1 l jyk p

=

px iX

il

pbi

py1

i q

Xy 1

j pl

Ý

/ /

il

p X X x ipy1 bi1r

pbiX

ls0

q

˜i X Ým qs 0 s i

x ipy 1 bi1r p

j pl

Ý

l

/

pbiX

ls0

p X bi1r

ks0 ls0

jp

Ý ls0

il q

px i

i

p X X x py1 bi1r l i

py1

bi1r p

s iXm˜ iX l

˜ iXy 1 Xq ŽÝ m qs 0 s i . s i

i

l

/

.

Repetition of this relation across si-conjugates of the first factor for fixed iX gives us py2 py1

žÝ Ý žÝ Ý žÝ Ý žÝ Ý ks0 ls0

py1

b1r p l l i jyk i q p px i

py2 py1

l jyk p

=

s

i q

py2 py1 l jyk p

Ý

j pl

q ˜ iX iy 1 Xm ŽÝ m qs 0 s i . s i

il

py1

i q l

pbiX

j pl

px iX

i q

Ý

i

pbi

py1 l

Xy 1

j pl

l

i

/ /

pbiX

iy1 q Ým qs 0 s i

l

p X X x ipy1 bi1r

ls0

q

˜i X Ým qs 0 s i

x ipy1 bi1r p

ls0

p X bi1r

ks0 ls0

Ý

/

p X X x ipy1 bi1r

ls0

px i

ks0 ls0

=

py1

bi1r p

l jyk p

pbi

l

px iX

py2 py1

jp

Ý ls0

p X bi1r

ks0 ls0

x py1 bi1r p l i

mi ˜ iXy 1 Xq ŽÝm qs 0 s i . s i

i

l

/

.

Now we may telescope the relations, as i runs from r to 1 y v , to yield py2 py1

r

Ł

is1y v

žÝ Ý žÝ Ý

l jyk p

bi1r p

ks0 ls0

py2 py1

=

ks0 ls0

l jyk p

px i

py1

i q

p X bi1r

px iX

l

Ý

j pl

x ipy1 bi1r p

ls0 py1

i q l

Ý ls0

j pl

pbi

r ms ˜ iX iy 1 q Xm ŽÝm qs 0 s i .Ž Ł ss iq 1 s s . s i

i

p X X x ipy1 bi1r

pbiX

l

/

i

l

Xy 1

q

˜i X Ým qs 0 s i

/

288

JOHN R. SWALLOW X

i y1

s

py2 py1

Ł

is1y v

ž

l jyk p

Ý Ý q

Ý

p X bi1r

l jyk p

žÝ Ý Ł žÝ Ý

px iX

ks0 ls0

X

q

/

py1

i q

Ý

l jyk p

bi1r p px i

p X X x ipy1 bi1r

pbiX

jp

i

/

r ms iy1 q ŽÝm qs 0 s i . Ł ss iq 1 s s

il

pbi

ls0

ms ˜ iXy 1 s Xq . Ł r X iX q m ŽÝm qs 0 i ss i q 1 s s

l

il

x py 1 bi1r p l i

Ý

j pl

ls0

ks0 ls0 py1

l

i

l

py2 py1

isi q1

r ms ˜ iX iy 1 q Xm ŽÝ m qs 0 s i .Ž Ł ss iq 1 s s . s i

pbi

py2 py1

r

il

x ipy 1 bi1r p

j pl

ls0

=

px i

ks0 ls0 py1

=

bi1r p

/

.

Telescoping again, with iX running from 1 y v to r, results in py2 py1

r

Ł

is1y v

l jyk p

žÝ Ý

ks0 ls0 py1

q

jp

Ý

Ł

i s1y v

ž

Ý

j pl

X

Ł

pbiX

py2 py1

r i s1y v

l jyk p

žÝ Ý

ks0 ls0 py1

q

il

p X X x ipy1 bi1r

ls0

s

/

il

px iX

ks0 ls0 py1

r ms rX ˜ iX iy 1 q Xm ŽÝ m qs 0 s i .Ž Ł ss iq 1 s s . Ł i s 1y v s i

p X bi1r

l jyk p

Ý Ý q

il

pbi

py2 py1

r X

px i

x py 1 bi1r p l i

ls0

=

bi1r p

Ý

r X m ˜ iXy 1 Xq ˜s ŽÝ m qs 0 s i . Ł ss i q 1 s s

i

l

/

p X bi1r

px iX

il

1 1r X x py biX p l i

jp

ls0

pbiX

m sq m ˜s ˜ iXy 1 s Xq . Ł r X iX q m ŽÝm qs 0 i ss i q 1 s s

il

/

.

Using this relation back in our cocycle formula, we find that t Ž ˜ g . t Ž g . g˜ equals r

Ł

is1y v

py2 py1

žÝ Ý

ks0 ls0

l jyk p

bi1r p px i

py1

i q l

Ý

j pl

ls0

hence the 1-cocycle condition is verified.

x ipy1 bi1r p pbi

p ?Ž m iqm ˜ i .rp @

i

l

/

t Ž gg˜. ;

p-EXTENSIONS

CENTRAL

289

OF GALOIS GROUPS

Using the conjugate-product relations of Theorem 4, the image of this cocycle t for Cases I and II, under the coboundary homomorphism, is

j p?Ž m 1y v qm˜ 1y v .r p@

r

Ł

X

is1y v

j pm˜ i m i .

The proof of Theorem 3 in wMa2x shows that the class of this 2-cocycle is precisely c. Finally, we show that the t for Case III satisfies the 1-cocycle condition and gives class c under the coboundary homomorphism. An inspection of the relations proved for Cases I and II yields that the t defined in Case III is also a 1-cocycle, for the cases are identical up to the introduction of another automorphism t 1 with similar properties to the si , and some multiplicative constants. Set g and ˜ g as before. The relations proved for Cases I and II, together with the different relation y s1y1 s yx 1t 1y1 , give us that t Ž ˜ g . t Ž g . g˜ equals

Ž y1.

X

m 1m ˜1

r

žŁŽ

y1 .

X

X

X

m i m iqm i m ˜ iqm ˜ im ˜i

is2

r

=

ž ž Ł

ž

2

/

/

/

2 bi1r2

y y y t 1 q yi q y t 1 i

2

2 ?Ž m 1 qm ˜ 1 .r2 @

2 ?Ž m iqm ˜ i .r2 @

x i q x is i q x i i y x is i i

is2

=

/ž

x 1 y x 1s 1 q x 1 i q x 1s 1 i

X

X

2 ?Ž m 1qm ˜ 1.r2 @

/

t Ž gg˜. .

The image of t under the coboundary homomorphism is then X

X

r

X

m m Ž y1. ?Ž m 1qm˜ 1 .r2 @q ?Ž m 1qm˜ 1.r2 @ Ł Ž y1. ˜ i i .

is1

Similarly, the proof of Theorem 3 in wMa2x shows that the class of this 2-cocycle is c. THEOREM 6. Let the notation be as in Theorem 5. The twisted algebra A˜ corresponding to the 1-cocycle t is a split p-cyclic K-algebra. Inside A m E it may be realized as Ž1, x . p , where for Cases I and II we ha¨ e

x s p Ž rq v .Ž py2. r

=

Ý

k 1y v , k 2y v , . . . , k rgZrpZ

=

Ý

k 1y v , k 2y v , . . . , k rgZrpZ

ž ž

ŽÝ Ł Ž xyp m bm .

k my1 s . r kl ss 0 s m Ł ls mq 1 s l

ms1y v r

Ý Ł Ž x mp bmy1 . Ž

ms1y v

k my1 s . r kl ss 0 s m Ł ls mq 1 s l

/ /

Ł ls 1 s l wpy2 r

kl

Ł ls 1 s l wpy1 . r

kl

290

JOHN R. SWALLOW

For Case III we ha¨ e that x is the product of

X k 1 gZr2Z

k1 , . . . , k r ,

r

=

ž

Ž Ž y1. k qk 1

Ý

Ł

ms1

Žsm xm

km

X 1

k 19

y Žt 1

X

w 0ŽŁ ls 1 s l

kl

.t 1k

y Ž1y t 1 . w 1ŽŁ ls 1 s l

kl

.t 1k

r kl k y1.Ž Ł ls mq 1 s l .t 1

y1.

X

1

r

1

.

/

and

k1 , . . . , k r ,

X k 1 gZr2Z

r

=

ž

Ž Ž y1. k qk 1

Ý

Ł

ms1

Ž1y s m xm

km

kX 1

X 1

r kl k .Ž Ł ls mq 1 s l .t 1

X

1

/

r

X

1

.

;

r sl r p4 and where wpy 2 , wpy1 g  Ł ls1y v al X s 1y v , s 2y v , . . . , s r g 0, 1, . . . , py14 for cases I r s r2 and II and w 0 , w 1 g ŽŁ ls1 a l l . b1s1 r2 4s1 , s 2 , . . . , s r , sX1 g 0, 14 for Case III.

Proof. We show that there exists a choice of wi in the theorem so that there exists an automorphism f of VA m E with t Ž g . s f g fy1, g g G, given by py1

i ¬ i,

j¬

l jyk p

py1

Ý ¨k Ý ks0

p

ls0

i l j,

where for Cases I and II, ¨ 0 s ¨ 1 s ??? s ¨ py3 s ¨ py 2 s

ž

Ý

k 1y v , k 2y v , . . . , k rgZrpZ

1 s p rq v ,

Ý

/

k 1y v , k 2y v , . . . , k rgZrpZ r

ž

Ł ls 1 s l wpy2

kl

ž

Ł ls 1 s l wpy1

kl

r

ŽÝ Ł Ž xyp m bm .

k my1 s . r kl ss 0 s m Ł ls mq 1 s l

ms1y v

/

,

and ¨ py 1 s

Ý

k 1y v , k 2y v , . . . , k rgZrpZ

r

r

Ý Ł Ž x mp bmy1 . Ž

k my1 s . r kl ss 0 s m Ł ls mq 1 s l

ms1

/

,

while for Case III, ¨ 0 and ¨ 1 are, respectively,

Ý k1, . . . , k r ,

X k 1 gZr2Z

Ž y1.

X

k 1 qk 1

kX 1

y Žt 1

y1.

w 0ŽŁ ls 1 s l r

kl

.t 1k

X

r 1

žŁ

ms1

Žsm xm

km

r kl k y1.Ž Ł ls mq 1 s l .t 1

X

1

/

p-EXTENSIONS

CENTRAL

291

OF GALOIS GROUPS

and

Ž y1.

Ý

X

kX 1

X

k 1 qk 1

y Ž1y t 1 . w 1ŽŁ ls 1 s l

k 1 , . . . , k r , k 1 gZr2Z r

=

ž

Ł

Ž1y s m xm

km

ms1

r kl k .Ž Ł ls mq 1 s l .t 1

X

1

r

/

kl

.t 1k

X q1 1

.

Then by Galois cohomology the algebra A˜ is realized as stated in the r theorem, with x s Ł 1y v¨ i. First, we claim that for any choice of wi in the theorem, the map f defined above is an E-homomorphism satisfying t Ž g . s f g fy1, g g G, where we view the 1-cocycle t as taking values in the automorphism group of VA m E via conjugation by the image. Note that it is an E-homomorphism because we may extend the definition on the generators to the whole algebra. We need only verify the latter condition. Recall that any sum of the form f s Ý g g G t Ž g .y1 Žm g ., where t Ž g . is interpreted via the conjugation-action as a linear transformation on the underlying vector space of A, will satisfy the t Ž g . s f g fy1 condition for m g A m E, since it is a Poincare ´ series wSe2x, Chap. X. We show that our f occurs in this fashion. Restrict f to the subspace of VA m E spanned by py 1 py 1  i l j4ls0 , which it maps homomorphically into  i l j4ls0 . Now, for Cases I and X r II, the conjugation action of t Ž g ., where g s Ł is1y v si m i Ł rjs1t jm j sends l jyk p

py1

Ý

p

ls0

l jyk p

py1

ilj ¬

Ý

p

ls0

ilj

for any k - p y 2, while for k s p y 2 and k s p y 1 we have, respectively, py1

Ý

py2.l jyŽ p

p

ls0

r

ilj ¬

ž

Ý Ł Ž x ip by1 i . Ž

m iy 1 s . r ms ss 0 s i Ł ss iq 1 s s

is1y v

py1

/

Ý

py2.l jyŽ p

p

ls0

ilj

and py1

Ý

py1.l jyŽ p

p

ls0

r

i j¬ l

ž

Ł Ž

is1y v

r ms iy 1 s ŽÝm ss 0 s i . Ł ss iq 1 s s xyp i bi

.

py1

/Ý

py1.l l jyŽ i j. p

ls0

For Case III, we have j q ij 2

¬ Ž y1 .

ž

X

m 1 qm 1

mX 1

y Žt 1

y1.

r

Ł x iŽ s

is1

i

mi

X

y1.Ž Ł rss iq 1 s sm s .t 1m 1

/

j q ij 2

292

JOHN R. SWALLOW

and j y ij 2

¬ Ž y1 .

ž

X

m 1 qm 1

mX 1

y Ž1y t 1

r

.

Ł x iŽ1y s

i

mi

X

.Ž Ł rss iq 1 s sk s .t 1m 1

is1

/

j y ij 2

.

py 3 py1 Ž yk l py 1 py1 Ž yk l l .l . . Let m s ŽÝ ks 0 Ý ls0 j p rp i j q Ý kspy2 Ý ls0 j p rp w k i j. Then the Poincare ´ series, as defined above, gives us precisely f, restricted to the subspace given. Now extend f as a linear transformation to all of A by extending f via algebra multiplication and linearity. Since the conjugation-action of f respects algebra multiplication and linearity, the condition f g fy1 s t Ž g ., g g G, then holds. Now we show that there must be some choice of Ž wpy 2 , wpy1 . which renders f an isomorphism. We consider Cases I and II only; Case III follows analogously. Counting dimensions, we find that the map f is an isomorphism of VA m E unless the image of j is a zero divisor. Raising the image of j to the pth power, we find that this occurs only if either

r

Ý

k 1y v , k 2y v , . . . , k rgZrpZ

ž

ŽÝ Ł Ž xyp m bm .

k my1 s . r kl ss 0 s m Ł ls mq 1 s l

ms1y v

/

Ł ls 1 s l wpy2 r

kl

/

Ł ls 1 s l wpy1

kl

or r

Ý

k 1y v , k 2y v , . . . , k rgZrpZ

ž

Ý Ł Ž x mp bmy1 . Ž

k my1 s . r kl ss 0 s m Ł ls mq 1 s l

ms1y v

r

is zero. We must choose wpy 2 and wpy1 so that these expressions are nonzero. We proceed by induction. If m s 1 y v , then we may choose p. wpy 2 , wpy1 g K Ž a1r so that m

Ý

k mgZrpZ

sm yp wpy 2 Ž x m bm . km

my1s s Ý kss 0 m

and

sm wpy1 Ž x mp bmy1 . km

Ý

k mgZrpZ

my1s s Ý kss 0 m

are nonzero. This follows from the fact that, multiplying the expressions by Ý ss 1 Ž pys. s m xm py 1

sy 1

and

Ý ss 1 Ž s s m . xm , py 1

s

Ý ss 0 s m respectively, and using the fact that x m s bm , we obtain py 1

s

Ý ss 1 Ž pys. s m p TrK Ž a1r wpy2 = x m m .r K py 1

ž

sy 1

and Ý ss 1 s s m p TrK Ž a1r wpy1 = x m . m .r K

ž

py 1

s

/

/

CENTRAL

p-EXTENSIONS

293

OF GALOIS GROUPS

p. If either were zero for all choices of wpy 2 , wpy1 in K Ž a1r , then by the m nondegeneracy of the trace form, x m s 0, contrary to hypothesis. Of p 2r p Ž py1.r p 4 course we need only consider wpy 2 , wpy1 in  1, a1r . m , am , . . . , am For the induction step, assume that for a given m g  1 y v , 2 y v , . . . , r y 14 we have that

m

c1 s

Ý

k 1y v , . . . , k m gZrpZ

ž

Ž Ł Ž xnyp bn . Ý

kny 1 s . m kl ss 0 sn Ł ls n q 1 s l

ns1y v

/

Ł ls 1y v s l wpy2

/

Ł ls 1y v s l wpy2

m

kl

and m

c2 s

Ý

k 1y v , . . . , k m gZrpZ

ž

Ý Ł Ž xnp bny1 . Ž

kny 1 s . m kl ss 0 sn Ł ls n q 1 s l

ns1y v

m

kl

are nonzero. We show that we may choose wpy 2 and wpy1 so that the above expressions are nonzero for m replaced by m q 1. The new expressions, multiplied by Ý ss 1 Ž pys. s mq 1 x mq 1 py 1

sy 1

and

Ý ss 1 Ž s s mq 1 . x mq1 , py 1

s

respectively, give us Ý ss 1 Ž pys. s mq 1 p 1r p 1r p 1r p wpy2 = x mq1 = c1 TrK Ž a1r 1y v , . . . , a mq 1 .r K Ž a1y v , . . . , a m . py 1

ž

sy 1

/

and Ý ss 1 s s mq 1 p 1r p 1r p 1r p TrK Ž a1r wpy1 = x mq1 = c2 , 1y v , . . . , a mq 1 .r K Ž a1y v , . . . , a m . py 1

ž

s

/

respectively. By the nondegeneracy of the trace, there must be some wpy 2 and wpy 1 , say a product of the previously chosen wpy2 and wpy1 by a p power of a1r mq 1 , so that our required expressions are nonzero. Therefore by induction wpy 2 and wpy1 exist. LEMMA 1.

A K-isomorphism h between A˜ ( Ž1, x . and A is gi¨ en by

i A˜ ¬ i A , py2 py1

j A˜ ¬

žÝ Ý

ks0 ls0

l jyk p

p

py1

i lA j A

/ žÝ qx

ls0

py1.l jyŽ p

p

/

i lA j A .

Proof. Clear. Using Theorem 2, we compute an expression for z and then take its reduced norm.

294

JOHN R. SWALLOW

THEOREM 7. Gi¨ en c / 0, a numerical decomposition of c as in Theorem 3, the existence of field elements of Theorem 4 corresponding to the sol¨ ability of c, and an appropriate choice of wpy 1 in Theorem 6, the complete set of solution fields to c is U

1rp

½ E Ž Ž kD . . : k g K 5 , where r

Ds

Ý

k 1y v , k 2y v , . . . , k rgZrpZ

ž

Ý Ł Ž x mp bmy1 . Ž

k my1 s . r kl ss 0 s m Ł ls mq 1 s l

ms1y v

/

Ł ls 1 s l wpy1 r

kl

in Cases I and II and D is

k1, . . . , k r ,

Ž Ž y1. k qk 1

Ý

X k 1 gZr2Z

X 1

kX 1

y Ž1y t 1 . w 1ŽŁ ls 1 s l r

kl

X

.t 1k 1

r

. ms1 Ł x mŽ1y s

ž

km r kl kX 1 m .Ž Ł ls mq 1 s l .t 1

in Case III. Proof. Using the expression for z in Theorem 2 and the maps f, h given in Theorem 6 and Lemma 1, we compute z to be py1 py1

p

Ý Ý

k jyk p

X

p

X

ks0 k s0

ž

py1yk kqly1

Ý ls0

Ł

msk

kq ly1 Ł msk ¨ m mod p

py1

¨ m mod p q

Ý lspyk

/

X X Ł mpy1 s0 ¨ m

X

ik .

The reduced norm gives py1

g s pr

Ł

ks0

ž

py1yk kqly1

Ý ls0

Ł

msk

py1

¨ m mod p q

Ý lspyk

kq ly1 Ł ms k ¨ m mod p X X Ł mpy1 s0 ¨ m

/

.

Substituting the values for the ¨ i determined in the proof of Theorem 6, g becomes p r Ž 1 q p rq v q p 2Ž rq v . q ??? qp Ž py2.Ž rq v . ¨ py 2 . = Ž 1 q p rq v q p 2Ž rq v . q ??? qp Ž py3.Ž rq v . ¨ py 2 q pyŽ rq v . . = Ž 1 q p rq v q p 2Ž rq v . q ??? qp Ž py4.Ž rq v . ¨ py 2 q pyŽ rq v . q py2Ž rq v . . = ??? = Ž 1 q ¨ py 2 q pyŽ rq v . q py2Ž rq v . q ??? qpyŽ py3.Ž rq v . .

ž

= 1q

1 p

Ž py2.Ž rq v .

¨ py2

q

1 p

Ž py3.Ž rq v .

¨ py2

q ??? q

1 p

rq v

¨ py 2

/

.

/

CENTRAL

p-EXTENSIONS

295

OF GALOIS GROUPS

Up to pth powers and scalar multiplication, g is 1r¨ py 2 , which, up to scalar multiplication again Žsince x g K U . is ¨ py 1. This gives us the D of the theorem.

5. CHOOSING wpy 1 Let c and a i be as in the notation of Theorem 3. Define the a i-component of the numerical decomposition of c to be the 2-cocycles in H 2 ŽŽZrpZ. n, Fp . given by ŽŽ a0 .. or Ž a i , bi ., i / 0, in Case I of Theorem 4; ŽŽ a1 .. q Ž a1 , b1 . or Ž a i , bi ., i / 1, in Case II; or Ž a i , bi ., i / 1, in Case III. When an a i-component of the numerical decomposition of the obstruction is itself split, we can simplify some of the factors of x in the algebra Ž1, x . of Theorem 6 and therefore of the solutions in Theorem 7. THEOREM 8. Suppose that an embedding problem c is sol¨ able and that the a i-component of a numerical decomposition of an obstruction is itself p tri¨ ial. Let the power of a1r which occurs as a factor in wpy1 as defined in i Theorem 6 and used in the determination of the solution in Theorem 7 be denoted l g  0, 1, . . . , p y 14 . The power l is determined as follows: Ži. If the a i-component is ŽŽ a i .., then l s 1. Žii. If the a i-component is Ž a i , bi ., then l s 0. Žiii. If the a i-component is ŽŽ a i .. q Ž a i , bi . and p / 2, then l s 1. Proof. If the a i-component is trivial, then the corresponding x i of p. Theorem 4 lies in K Ž a1r ; hence all sj , j / i, leave x i invariant. i By Theorem 4, then, si leaves x j invariant for j / i. In the induction step of Theorem 6, then, when m s i, we have that the c 2 determined thus far, for smaller m, is nonzero and si-invariant. By the proof of Theorem 6, we must choose an l such that, with wpy 1 depending on l as given in Theorem 6, y1

si p 1r p p 1r p TrK Ž a1r wpy1 = x iÝ ss 1 s s i .r K Ž a1r 1y v , . . . , a i 1y v , . . . , a iy 1 .

ž

py 1

sy 1

= c2

/

is nonzero. Now by the proof of Theorem 7, the D of Theorem 7 for the embedding problem described by the a i-component is 1 sy1 i

wpy 1 =

py 1 sy 1 x iÝ ss 1 s s i

y1

si p 1r p p 1r p TrK Ž a1r wpy1 = x iÝ ss 1 s s i .r K Ž a1r 1y v , . . . , a i 1y v , . . . , a iy 1 .

ž

py 1

y1

sy 1

/.

si Hence for any wpy 1 such that the trace is nonzero, then wpy1 = x iÝ ss 1 s s i is a solution to the embedding problem. py 1 sy 1 We know, however, from wMa2x, Theorem 3, that a li r p x iÝ ss 0 s s i is a solution to the embedding problem described by the a i-component, and it py 1

sy 1

296

JOHN R. SWALLOW

is easy to show that if l is replaced by a different value, it is not possible that both can be solutions to same the embedding problem. Therefore the trace must be zero for all choices of l save the one from wMa2x, Theorem 3, and must be nonzero for the correct l. In the induction step of Theorem p 6, then, the necessary power of a1r , required to alter wpy1 from the step i lr p before, is a i . The values of l from wMa2x, Theorem 3, are given in the statement of the theorem and we are done. Remark. A similar statement is unavailable for the ŽŽ a1 b1 .. q Ž a1 , b1 . component of Case III, since the fact that the 2-cocycle is split is not enough information to determine a value for wpy 1 which always insures that D is nonzero, for any choice of x 1 and y satisfying the conditions of Case III. Compare wMa2x, Theorem 3ŽB.Ž28..

6. EXAMPLES EXAMPLE 1. The Dedekind]Witt example ŽCase III; cf. wDex, p. 379; wWix, p. 245; wMa2x, Example 5.. The overfields of QŽ'2 , '3 . which are quaternion over Q are the fields U

½ Q ž 'k Ž6 q 3'2 q 2'3 q 2'6 . / : k g Q 5 . To derive this result, we set p s 2, K s Q, E s QŽ'2 , '3 .. A numerical decomposition of the obstruction is Ž6, y1. q Ž2, 3., using Theorem 3. We have elements x 1 s Ž1 q '2 .Ž'2 q '3 . and y s '2 q '3 satisfying the conditions of Theorem 4, with '2 s1y1 s '3 t 1y1 s y1 and '3 s 1y1 s '2 t 1y1 s 1. Using Theorem 6 with an initial guess of w 0 s w 1 s 1, we find that

x s ¨ 0¨ 1 s 96, where ¨ 0 s Ž 1 y y t 1y1 y x 1s1y1 q y t 1y1 x Ž s 1y1 .t .

s Ž 1 y Ž y5 q 2'6 . y Ž y3 q 2'2 . q Ž 15 y 10'2 y 6'6 q 8'3 . . and ¨ 1 s Ž 1 y y 1y t 1 y x 11y s 1 q y 1y t 1 x Ž1y s 1 .t .

s Ž 1 y Ž y5 y 2'6 . y Ž y3 y 2'2 . q Ž 15 q 10'2 q 6'6 q 8'3 . . . Note that ¨ 1 s 4)Ž6 q 2'3 q 3'2 q 2'6 .; with Theorem 7, this gives the result.

CENTRAL

p-EXTENSIONS

297

OF GALOIS GROUPS

EXAMPLE 2. Case I, p s 2 Žcf. wMa2x, Example 1Ži., Example 6.. Set p s 2. Let K be a field containing 'y 7 such that 1, y1, 2, and 5 are linearly independent classes of K U rK U 2 Žfor instance, Q 2 .. Let E s K Ž'y 1 , '2 , '5 .. Let a0 s y1, a1 s 2, and b1 s 5; let s 0 , s 1 , t 1 be the nontrivial automorphisms of K Ž a0 ., K Ž a1 ., K Ž b1 ., respectively, extended trivially to the others. The obstruction to lifting E to a 2-extension of E, which is dihedral over K Ž a0 . with the lifts of s 1 and t 1 of order 2 and cyclic over K Ž a1 , b1 ., can be numerically decomposed as ŽŽy1.. q Ž2, 5.. Elements satisfying the conditions of Theorem 4 are then

'

'

'

'

' '

x0 s

'2 Ž '2 q '7 . , Ž 1 q 'y 1 .Ž 2 q 'y 1 .

x 1 s '2 q '7 .

We guess w 0 s w 1 s 1 as in the previous example, and we compute ¨ 0 s 1 q x 1s1y1 y x 0s 0y1 y x 0Ž s 0y1 . s 1 x 1s1y1

s 1q

ž

9 y 2'14 5

/ž

1q

2'y 1 Ž 3 q 4'y 1 . 10

/

and ¨ 1 s 1 q x 11y s 1 y x 01y s 0 y x 0Ž1y s 0 . s 1 x 11y s 1

s 1q

ž

9 q 2'14 5

/ž

1q

y2'y 1 Ž 3 y 4'y 1 . 10

/

.

The x of Theorem 6 is then 56r25. The set of solution fields is, by Theorem 7,  K ŽŽ k¨ 1 .1r2 : k g K U .4 . Note that ¨1 s

y'y 7 Ž 3 q 'y 1 . 125

2

Ž '2 q '7 .Ž 3 y 'y 1 . .

Hence our result agrees with wMa2x, Example 6.

REFERENCES wCr1x wCr2x wCr3x

T. Crespo, Explicit construction of A˜n type fields, J. Algebra 127 Ž1989., 452]461; erratum, J. Algebra 157 Ž1993., 283. T. Crespo, Explicit solutions to embedding problems associated to orthogonal Galois representations, J. Reine Angew. Math. 409 Ž1990., 180]189. T. Crespo, Embedding Galois problems and reduced norms, Proc. Amer. Math. Soc. 112 Ž1991., 637]639.

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wDaMrx P. Damey and J. Martinet, Plongement d’une extension quadratique dans une extension quaternionienne, J. Reine Angew. Math. 262 r 263 Ž1973., 323]338. wDaPx P. Damey and J.-J. Payan, Existence et construction des extensions Galoisiennes et non-abeliennes de degre differente de 2, J. Reine ´ ´ 8 d’un corps de caracteristique ´ ´ Angew. Math. 244 Ž1970., 37]54. wDex R. Dedekind, ‘‘Konstruktion von Quaternionkorpern,’’ Gesammelte Mathematische ¨ Werke, Band 2, Vieweg, Braunschweig, 1931, pp. 376]384. wFrx A. Frohlich, Orthogonal representations of Galois groups, Stiefel]Whitney classes ¨ and Hasse]Witt invariants, J. Reine Angew. Math. 360 Ž1985., 84]123. wGSSx H. Grundman, T. Smith, and J. Swallow, Groups of order 16 as Galois groups, Exposition. Math. 13 Ž1995., 289]319. wHox K. Hoeschmann, Zum Einbettungsproblem, J. Reine Angew. Math. 229 Ž1968., 81]106. wMa1x R. Massy, Solutions explicites de problemes de plongement, J. Number Theory 20 ` Ž1985., 299]314. wMa2x R. Massy, Construction de p-extensions Galoisiennes d’un corps de caracteristique ´ differente de p, J. Algebra 109 Ž1987., 508]535. ´ wMaNx R. Massy and T. Nguyen-Quang-Do, Plongement d’une extension de degre ´ p 2 dans une surextension non abelienne de degre ´ ´ p 3 : ´etude locale-globale, J. Reine Angew. Math. 291 Ž1977., 149]161. wSe1x J.-P. Serre, L’invariant de Witt de la forme TrŽ x 2 ., Comment. Math. Hel¨ . 59 Ž1984., 651]676. wSe2x J.-P. Serre, ‘‘Corps Locaux,’’ Hermann, Paris, 1962. wSw1x J. Swallow, Constructive solutions to central embedding problems, Ph.D. Dissertation, Yale University, May 1994. wSw2x J. Swallow, Embedding problems and the C16 ª C8 obstruction. Recent Developments in the Inverse Galois Problem, Contemp. Math. 186 Ž1995., 75]90. wSw3x J. Swallow, Solutions to central embedding problems are constructible, J. Algebra 184 Ž1996., 1041]1051. wWix E. Witt, Konstruktion von galoisschen Korpern der Charakteristik p zu ¨ vorgegebener Gruppe der Ordnung p f , J. Reine Angew. Math. 174 Ž1936., 237]245.