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ON THE GALOIS MODULE STRUCTURE OVER CM-FIELDS. Jan Brinkhuis. In this paper we make a contribution to the problem of the existence of a normal ...
manuscripta math.

75, 3 3 3 -

347

manuscripta mathematica

(1992)

9 Springer-Verlag 1992

ON THE GALOIS MODULE STRUCTURE OVER CM-FIELDS

Jan Brinkhuis

In this paper we make a contribution to the problem of the existence of a normal integral basis. Our main result is t h a t unramified realizations of a given finite abelian group A as a Calois group Gal(N/K) of an extension N of a given CM-field K are invariant under the involution on the set of all realizations of A over K which is induced by complex conjugation on K and by inversion on A. We give various implications of this result. For example, we show t h a t the tame realizations of a finite abelian group A of odd order over a totally real number field K are completely characterized by ramification and Galois module structure. Introduction By a classical theorem of Hermite each algebraic number field

K

degree

has and

refinement question.

only

finitely

discriminant of

To

this what

many

over

result extent

field

K.

In

one are

extensions

an

attempt

could the

ask

a

following

of

a

Gal(N/K) of

finite abelian group A as a Calois group

given

towards

the

realizations

of

given a tame

field extension N of a given number field K characterized by their

'ramification'

where

the

latter

and

their

is defined to

'Galois be

the

module

structure',

isomorphism class

of

on, the ring of integers in N, as a module over the group ring onA of A over OK? It has been shown in [3] t h a t this question

is

equivalent

to

the 333

following

one,

which

seems

BRINKHUIS

more

restricted

which

are

if

a

is

a

sight:

-at

normal

paper

K

first

unramified

moreover present

at

are

all finite

integral

we o f f e r

primes-

basis

the

CM-field

realizations

over

and K,

following result

(our

definition

of A

of

over

K

which have

rare?

In

the

on this

question

CM-fields

includes

totally real fields).

All unramified integral

basis

realizations

are

of

invariant

,d over K with a normal

under

on the set of all realizations

of A

the

'obvious'

involution

over K which is induced

by complex conjugation on K and by inversion on ,4. In the remainder consequences real,

the

abelian

of

this

result

a

normal

exception

of

exceptions implies

be

of

the

case

restated

as

that

number

integral

basis

over

K,

of

indeed

free

quadratic

exist, ones.

as The

K

with

will

Galois

K

the of

see.

module

totally

unramified

field

extensions

we

is

follows:

real

following

locally

the

totally

an unramified realization of

In

we point out some

a

composita

can

the

result.

can

extensions

never

of this introduction

have

possible K.

Such

This

result

structure

of

of A over K, viewed as an element

class

group

Cg(oK'4)

has

order

either

exp,4 or e x2 p A where exp'4 is the exponent of the group "4. Thus,

in

particular,

odd.

Moreover,

beginning

of

realizations totally

order

returning this

to

is precisely the

question

the

introduction,

exp,4

tame

if

1,41 is

posed

at

the

Galois

algebra

of a finite abelian group ,4 of odd order over a

real

ramification

this

number and

field

Galois

K

are

module

completely structure.

such K and ,4, the unramified realizations

characterized

by

particular,

for

In

of "4 over K have

mutually non-isomorphic Galois module structures. In the other case, nmnely that K is a totally imaginary quadratic

extension

of

a

totally 334

real

number

field,

our

BRINKHUIS

result

has for

Hilbert

if IlK, the

example the following consequence:

class field

of K,

has

a normal integral

basis

over

K, then hK+=l or 2, where hK+ is the class number of K +, the maximal real subfield of K. via

Taking

a

different

Galois

modules

point

of

non-trivial

existence of which cannot

-at

view,

our

elements

result

in

least so f a r -

exhibits

Cs

the

be demonstrated

otherwise. Finally we remark that the old problem of the existence of normal

integral

bases has led in the last twenty years to

an

extensive

literature

of

integers,

with

on

powerful

however

it

seems

that

possible

to

obtain

precise

structure basis

and

in

problem

Galois

methods

this

is

on

the

extensions

the inside

structure

and

on

classical

rings

theorems;

time the

the

of

deep

first

information

particular

of

module

that

it

is

Galois module normal

Hilbert

integral

class

field

of a CM-field.

1.

T h e main r e s u l t Let

rational

QC

of

QC.

embedding

with

complex

requirement totally

number

algebraic

of

Q,

the

field

fields will be considered

Let

a

K

be

CM-field,

an automorphism

that which

is,

of

to

it

be

is

coincides

a for

of K into C, the field of complex numbers, conjugation.

that

This

K is either

imaginary field.

closure

all number

field which has

each

a

an

numbers;

subfields number

be

In

a totally

quadratic the

last

is

extension case

we

CM-field. Let A be a finite abelian group.

335

equivalent real of speak

to

number a

field

totally of

the

a

or real

proper

BRINKHUIS

(1.1)

Definition.

A realization

of

A over

K is a

pair

r=(N,~) consisting of a Galois extension N of K together with an isomorphism ~ from Gal(N/K) to A. Now we choose an embedding of QC into C and we restrict complex conjugation on C to an automorphism

o f QC which we

denote b y c. For each group G and each G-module A we will denote the action of G on A by the left exponential notation

(g,a) -~ ga exponential

(for

geG,

all

notation

more

aeA).

We

generally

to

will

use

the

denote

the

action

left of

the group ring ZG on A. (1.2) K,

D e f i n i t i o n . For each realization r = ( N , ~ ) its

of ,5 over

complex conjugate realization -r=(N,~) is defined as

follows

N= {CnlneN} ~b(w)= r

-1 for all wEGal(N/K)

Warning. Notice the second inverse sign. Inversion

(1.3)

on the group A 'plays the role of complex conjugation'.

Remark. This definition

(1.4)

does

not

depend

on

the

chosen embedding of QC into C.

A realization unramified said

to

at have

all a

r=(N,~)

is

called

unramified

finite

primes.

A

realization

normal

integral

basis

if

if

N/K is

r=(N,~b)

ON, the

ring

is of

integers in N, is a free module on one generator on OKA, the group ring of A over o K.

336

BRINKHUIS

Now we come to the main result of this paper.

(1.5)

Each

Theorem.

unramified

abelian group over a CM-field

realization

of

a finite

with a normal integral basis

is equal to its complex conjugate realization.

This

result

can

also

be

stated

as

follows.

Let

K + be

the

maximal real subfield of K.

Each unramified abelian extension N of a CM-field

(1.6)

K with a normal integral basis over K is Galois over K + and moreover

the

action

of

complex

conjugation

on

Gal(N/K)

by

conjugation in the group Gal(N/K +) is equal to inversion.

In

a

2.9-

previous and

theorem;

in

paper

[4] a

[2] we proved

certain

related a

corollaries

result

much of

is given

weaker

theorem

-theorem

version

(1.5),

of

this

to

be

given

the

following

below, have been obtained already in [2] and [4].

Our

proof

criterion.

of theorem Before

preparations.

We

(1.5)

stating recall

will be it,

we

that

based

on

have

to

Galois

make

theory

some

establishes

a

bijection between the group HI(/2K, A)=Hom(/2K, A), the set of all continuous homomorphisms r from $2K=Cal(QC/K) to .4, and the set of Galois algebras A over K with Galois group A. For each q~eHom(~2K,A ) let Ar be the corresponding Galois algebra and let K s be (Qc)KerS, the fixed field of Kerr of r

One says that

As

is unramified

KS/K

is

at

all

unramified

finite

if the

primes.

the

kernel

field extension

Let

as

be

the

maximal order in As. One says that As has a normal integral basis if a s -~ o/(A as oKA-modules. We consider the action of 337

BRINKHUIS

the group 12K on the ring ?]CA - t h e group ring of A over ?]c the

ring

on

A (c_?]CA) and which acts

action. into

of

The

algebraic

in QC -

the

group

f2K-action

and

which acts

on ?]c (c_TCA) by

inclusion map i from A

?]CA*,

preserves

integers

(with trivial

of

invertible

elements

so

it

a

induces

trivially

the Calois OK-action)

map

in

?]CA,

of

GMois

cohomology groups i•

Hom(J2K, A)=HX(12K,AI) -~ Hx(S2K,TCA*)

Now we can state

the promised criterion (for

the proof we

r e f e r to [4]).

(1.7)

Let

Proposition.

The

r e Horn(OK, A).

following

conditions on r are equivalent (i )

The

Galois

algebra

A~

corresponding

to

r

is

role

in

unramified and has a normal integral basis. (ii) Moreover

i•162 = 1. the

following

result

will

play

a

crucial

our proof. Let G be the product group OK+ x C 2 and let a be the n o n - t r i v i a l

element of Cz. We will consider 12K+ and C2

as subgroups of G. We let the group G act on the group ring ZCA as follows: let 12g act on Z c by the Galois actions, on A trivially,

let a

act on A by inversion and Z c trivially. The

action of the element c ~ = ( c , a ) e G complex

conjugation.

Let

#

be

unity in QC.

338

on ZCA plays the role of the

group

of

all

roots

of

BRINKHUIS

P r o p o s i t i o n . Let ueT?CA * and assume that u m e o K A * f o r

(1.8)

same m > 1. T h e n

1-(~UE #.A To begin with, (x acts on the field components of the

Proof.

semisimple algebra

KA, which are

conjugation. Therefore,

all CM-fields,

as complex

if we project the element 1-~(um) of

KA on one of the field components of KA, then we get an algebraic that

unit

such

l-~(um)

of

units

and

absolute must

value

1; it

roots

of

be

so 1-~u is a

torsion

is a

unity.

standard It

result

follows

that

element in ZCA*. Finally,

it is well-known that the torsion subgroup of ZCA* is #.A. [] *

We write w = c

-1

wc for

all w~S2 K. We define for each

CeHom(~2K, A ) its 'complex conjugate' (~eHom(S2K, A ) as follows -1 for all we/2g.

r162

Now

we

are

ready

to

state

and

prove

the

following

implication which is the key step in the proof.

(1.9)

Proof.

Proposition.

Assume i•162

CeHom(f2K, gl ).

Let

This means that

If

i•162

then

there is an element

u e Z C A * such that

(1.10)

~(w ) = W-lu

Let m be the order of r

V0,)e ~K

Then, by (1.10), u

and so, as (;~cA*)S2K=oKZ1*, 339

m

is fixed by

~2K,

BRINKttUIS

(1.11)

u m e o//A*.

Therefore,

by

proposition

(1.8),

1-C~UE#'A

and

so

we

can

write (:.:2)

1-% = ~p

with ~ e # and p e A. Now we are going to compute for all weE2 K the element

(r

0 =

in

two

different

equal to ~q(~p),

ways.

this

equal last

the

one

hand,

by

(1.12),

0

is

which equals ~,-l~e#. On the other hand, 0

is equal to ~-lu-(~-l)au, (1.10),

On

so, as (w-1)c~=c~(w*-l), this is, by

to r162

expression

as c~ acts is

equal

to

on A by

O(w)r

Comparing

these two outcomes and projecting from # . A = # x A r162

onto A we get

= 1.

This holds for all weE2K, t h a t is, r 1 6 2 By proposition

inversion,

[]

(1.7) and proposition

(1.9) we have

now

p r o v e d the following result. T h e o r e m . Let CeHom(f2r, A ). I f the Galois algebra AV

(1.5)'

corresponding to r

is unramified and has a normal integral

basis, then r = r We have seen

to

be

also just

proved

theorem

theorem

(1.5)' 340

(1.5)

as

with

this

is readily

the

additional

BRINKttUIS

assumption that r is surjective. 2.

Consequences We

start

with

the

implications

of

our

result

normal integral basis problem. These are r a t h e r

for

strong

the if K

is totally real.

(2.1)

Corollary.

extension

of

integral

any

basis

There

totally

with

the

is

real

no

unramified

number

possible

field

exception

with of

abelian a

normal

composita

of

quadratic extensions. Proof. This is just a reformulation of theorem (1.5) for the case that

K is totally

real. To

show this,

it

is convenient

to use the terminology of theorem (1.5)' let eeHom(/2K,A ) be such

that

the

Galois

algebra

AV is

unramified

and

has

a

normal integral basis. To establish the corollary we have to show r

As K is totally real, the automorphism c lies in

~2K and so, as r that

r162162

is ~ = r

~b=~,

Therefore

the

comes down to r

for all ~e~2/~,

conclusion

of

theorem

(1.5)',

This finishes the proof of the

corollary. []

The

possible

exceptions mentioned in (2.1)

can actually

occur for certain K. For example let K be a real quadratic number there

field are

of

odd

precisely

discriminant.

2k - l - 1

It

unramified

is

well-known

quadratic

that

extensions

of K where k is the number of prime numbers dividing the discriminant (see

of

K.

In

proposition (IV 3.5a)

[1] in

we

proved

[6]). 34]

Let

the

S(K)

following be

the

result set

of

BRINKHUIS

number

fields

which

are

composita

of

unramified quadratic

extensions of K. (2.2)

T h e o r e m . Let K be a real quadratic number field of

odd discriminant.

I f the norm of the fundamental unit equals

+1,

is

then

integral If

there

basis

a

unique

over K;

this norm is

-1,

this

field is

N e S( K)

with

a quadratic

then no N e S ( K )

a

normal

extension

of

K.

has a normal integral

basis over K. Now assume that K is a proper CM-field. By class field theory the Galois group over K of HK, the maximal unramified abelian the

extension

class

of

group

K,

of

unramified abelian

is K.

canonically

isomorphic

Therefore,

by

Galois

extensions of K correspond to

to

Cr

theory, subgroups

of CQ~. We recall that the norm map from Cr K to CeK+ is surjective

and

C~K+ to Cr

(2.3)

that

the

the

canonical map

C M - field

K

has

a

normal

integral

basis

the

Let eeHom(~2K, A ) correspond to an unramified Galois

a canonical way over a homomorphism r Translating the

the

then

contains the image of C~K..

algebra As with a normal integral basis. Now r

of r

from

an unramified abelian extension of a

corresponding subgroup of Cr

Proof.

of

has either i or 2 elements.

Corollary. I f

proper

kernel

conclusion of theorem

we get r162 action

of

complex

factorizes in

from CeK to

(1.5)',

r162

A.

in terms

-1 for all xeCQ~, where - denotes conjugation

on

CdK.

Therefore

r

By the remarks preceding the corollary it follows

that Ker r

contains the image of Cdn+. [3 342

BR/NKHUIS

In particular,

letting

hK+ be

the

class

number

of K +,

we get the following result.

(2.4)

If

Corollary.

HK,

the

Hilbert

class

field

of

a

proper C M - f i e l d K, has a normal integral basis over K, then hg+----1 or 2. Proof.

Immediate

from

corollary

(2.3)

and

the

remarks

preceding it. []

For

unramified abelian

extensions N/K,

the

oKA-module

oN - w h e r e AI=Gal(N/K)- is locally free (or, what is the same here,

projective)

class

can

be

of

rank

viewed

as

an

one.

Therefore

element

of

its

the

isomorphism

following finite

abelian group, Cg(oKA), the class group of the ring ogA. It is trivial basis.

precisely

The

if

results

the

above

element is non-trivial,

extension show

that

has in

so one is naturally

a

normal many

integral

cases

led to

this

ask what

its order in the group Cg(OKA ) is. We will denote this order by

ord(o N).

We

can

give

the

following

divisibility

result

for it.

(2.5)

Corollary.

Let K be a CM-field

and A

a finite

abelian group. Let an unramified realization Gal(N/K) .~ A of A over K be given and let r be the corresponding element in Hom(~2n, A). Then ord(0r

[ord(ON) [ordr

Proof. Let H} be the maximal unramified abelian extension of K. We will view Hom(Gal(H~/K),A) as a subgroup of Hom(~2K, A ) via

inflation.

Let g be

the map 343

from ttom(Gal(H~/K),A)

to

BRINKHUIS

Cd(oKA) which sends each r

to the o/r

class of

ere, the maximal order of the Galois algebra As. We have to p r o v e that, for all subjective r (2.6) This

one has

ord(r162 Iordg(r holds

in

fact

more

ordr

generally

without

the

surjectivity

condition: it is known that g is a homomorphism, moreover we have p r o v e d that (2.6)

is an

the condition g ( r

elementary

group

implies r 1 6 2

theoretic

consequence

clearly of

these

facts. [] If

K

is

totally

real,

this

result

amounts

to

the

following one. (2.7)

I f N is an unramified abelian extension

Corollary.

of a totally real number field K, then ord(oN) is k or v2k where k

is

the

exponent

of

the Galois group

of N/K.

In

particular, if [N:K] is odd, then o r d ( o g ) = k . Finally beginning

we

turn

of the

to

introduction

the following definitions. a

finite

abelian

called tame F,

the

if for

to

mentioned

this paper.

In

at

the

[3] we made

F/E of number field is

An extension

each

finite prime

index

is

not

A tame realization

b e a pair (M,r together

question

Let K be a CM-field and let A be

group.

ramification

characteristic.

the

of E which ramifies

divisible

by

the

in

residual

of A over K is defined to

consisting of a tame Galois extension M of K

with an isomorphism from Cal(M/K) to A. Two tame

realizations

(M,r

and

{N,~p)

of

A over

K are

defined

to

h a v e the same ramification if they become isomorphic under a suitable degree,

unramified

base

field

extension

L

of

K

of

in the following sense: the tensor products L| 344

finite and

BRINKHUIS

L|

N are

isomorphic

as L - A - a l g e b r a s ,

that

is, there

isomorphism of L-algebras from LOKM to L| A - a c t i o n s . Two tame realizations (M,r are

said

to

have

the

same

Galois

is an

which preserves

and (N,~b) of A over K module

structure

if

the

maximal orders OM and oN are isomorphic as modules over the group ring o/cA. We proved the following result in [3].

(2.8)

Two tame realizations (M,r

have

the

same

ramification

and

and (N,~p) of A over K the

same

Galois

module

structure if and only if i•162215 ,•

Here ~

is as

in proposition

and ~b with the

(1.7) and

we have

identified

r

elements of Hom(~K,ZI) which one gets from

them b y inflation. The over

K

question, are

module

to

what

characterized

structure

is

extent by

tame

their

therefore

realizations

ramification

essentially

of

and

equivalent

A

Galois to

the

question how far the homomorphism i • is removed from being injective. implies

The

result

r162

gives

of

the

present

information

in

a

paper positive

that

i•162

direction.

For

example, we thus get the following result.

(2.9)

Corollary.

Let

and let Zl be a finite tame

realizations

of

K

be

a

totally

abelian group A

over

K

real

number

of odd order.

are

field

Then the

characterized

by

their

ramification and their Galois module structure.

Proof. Let (M,r K

with

structure.

the

and (N,~P0) be tame realizations of A over

same

Let r

we get from r

ramification

and

the

same

Galois

module

resp. ~p be the element of Hom(S2K, A ) which resp. ~P0 by inflation. 345

Then b y (2.8) we get

BRINKHU/S

i•162 = i•

and

proposition (1.9),

so

i•162 -1) = 1.

r162

by

AS K is totally real, this is

equivalent to the condition (r order, r

Therefore,

that is r

and so, as A has odd

This proves (M,r

[]

In particular we get the following result. (2.10) and

Corollary. A

a

unramified

finite

Let

K

abelian

realizations

be a group

of

A

totally of over

real

odd

number

order.

Then

field the

K

have

mutually

and

non-isomorphic Galois module structures.

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[1]

J. Brinkhuis, Embedding problems doctoral dissertation, Leiden (1981)

Galois

modules,

[2]

J. Brinkhuis, Normal integral bases and conjugation, J. reine angew. Math. 375/376, (1987)

complex 157-166

[3]

J. Brinkhuis, Galois module structure obstruction to a local-global principle, to the Journal of Algebra

[4]

J. Brinkhuis, C M - f i e l d s and (1989)

[5]

L. Childs, The group of unramified Kummer extensions of prime degree, Proc. L.M.S. 35/3, 407-422 (1977)

[6]

A. Fr6hlich, Galois module structure integers, Ergebnisse der Math. 3, 1 (1983)

[7]

V. Fleckinger et T. Nguyen Quang Do, Bases nor'males, unitds et conjecture faible de Leopoldt, preprint (1990)

as the appear in

Unramified abelian extensions of their Galois module structure, preprint

346

of

algebraic

BRINKHUIS

[8]

I.

Kersten

CM-fields,

J.

and J. Michalis Number Theory,

Zp- extensions 32,

no.

2,

of

131-150

(1989)

[9]

Kersten and J. Michaligek: On Vandiver's conjecture and 7p-extensions of Q(~pn), J. Number Theory, 32, no.

I.

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Jan Brinkhuis Econometric Institute Erasmus University P.O. Box 1738 3000 DR Rotterdam The Netherlands

(Received February 7, 1991; in revised form January 23, 1992)

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