A note on free pro-$p$-extensions of algebraic number fields

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

M

ASAKAZU

Y

AMAGISHI

A note on free pro-p-extensions of algebraic number fields

Journal de Théorie des Nombres de Bordeaux, tome

5, n

o

_{1 (1993),}

p. 165-178

<http://www.numdam.org/item?id=JTNB_1993__5_1_165_0>

© Université Bordeaux 1, 1993, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

A

note on

free

pro-p-extensions

of

_algebraic

number fields

par MASAKAZU YAMAGISHI

ABSTRACT. For an _algebraic number field k and a _{prime p,} define the

num-ber _{p to} be the maximal number d such that there exists a Galois extension

of k whose Galois _{group is} a free _{pro-p-group of rank d.} The _Leopoldt

con-jecture implies 1~ _{p ~} r2 +1, (r2 denotes the number of _{complex places} of

k). Some _examples of k and _p with _03C1 = _{r2 + 1} have been known so far. In this _note, the invariant 03C1 is studied, and _{among other things} some _examples

with p r2 + 1 are _given.

Introduction

In this note we shall consider free

_{pro-p-extensions}

(i.e.

Galois

exten-sions whose Galois groups are free

pro-p-groups)

of

algebraic

number fields.

This is a natural

_{generalization}

of

_{Zp-extensions}

since

_Zp

is a free

pro-p-group of rank one. It would be of interest to

_generalize

some

deep

results

in Iwasawa’s

_theory

on

_{Zp-extensions}

to the case of free

pro-p-extensions

of

_arbitrary

ranks.

_(For

local

_fields,

some results have been obtained

by

T.

_Nguyen

_Quang

Do

_[11,

_§8].)

Our

_guiding

_problem

in this _note,

_however,

is to determine the maximal rank _p of free _pro-p Galois _groups over a fixed

_algebraic

number field k for a fixed

_prime

_p. If the

_{Leopoldt conjecture}

is true for k and _p, then

by

class field

_theory

we have _p r2 +

_1,

where r2 denotes the number of

complex places

of k

_{(cf. (1.5)}

_below).

Some

_examples

of k and _p for which the

_equality

_p = _{r2 +} 1 hold _are known. The main results of this _{note are}

the

_following:

(1)

We shall _prove

_p

_r2+1

under the

_assumption

that the "weak

_Leopoldt

conjecture"

is true for all

_{Zp-extensions}

of k

_(Proposition

_3.5).

(2)

If _p is

_odd,

k contains a

primitive

p-th

root of

_unity,

and if there exists

Mots-clés : _Algebraic number _{field, Zp-extension, free pro-p-group.} Manuscrit requ le 22 _septembre 1992, version definitive le 31 _juillet 1993.

a

prime

vo of k which does not

decompose

at all in the maximal pro-p-extension of k unramified outside _p, then _{p is}

_{explicitly given by}

where kv

denotes the

_completion

of k at v. This is a

special

case of

Corollary

4.6.

_Using

this

_formula,

we shall

_{give examples}

of k

and p

such

that p

r2 + 1 is a strict

_inequality.

For the

_proof

of

_(2),

we use K.

_Wingberg’s

"free

_{product decomposition"}

of Galois _groups

_{([20], [21]).}

The

_assumption

of the existence of vo in

(2)

seems

_quite

_strong.

For

example

this

_implies

the

_validity

of the

_{Leopoldt conjecture}

for k and _p.

We think that we are far from the

_complete

determination of _p in

_general.

Acknowledgement.

The author wishes to _express his

_hearty

thanks to Professor Shoichi

_Nakajima,

under whose

_guidance

this work was

done,

to

Professor Mamoru Asada for the

_proof

of Lemma

_2.1,

and to other members of the Number

_Theory

Seminar at

_Komaba,

_Tokyo,

_especially

to Professor Kenkichi

_Iwasawa,

for valuable advice and comments. The author is also

grateful

to the referee for useful comments on the

bibliography.

1. Formulation of the

_problem

Free pro-p-groups

(cf. [15]).

Let _p be a

_prime,

which will be fixed

throughout

this note. _{For any} set I of

_indices,

the free _pro-p-group

_F(I)

generated by

is defined

_{([15,1-1.5]),}

and

_F(J)

if and

_only

if

j(7

=

#J,

(~

denotes the

cardinality

of _a

set).

The

cardinality

of I is called

the rank of

_F(I),

which we denote

by

rk(F(I)). (For

a

general

_pro-p-group

G,

rk(G)

is defined to be the

_cardinality

of a, minimal

_generating

subset of G’. This is

_equal

to

_dimz/pz

cf.

_{[ 15,}

_{1-4.2] .)}

In all cases of interest _{to us,} the rank will be finite. We shall write

_Fd

instead of

_F(I)

when d =

_~I

is finite. In

_particular,

Fl

=

Zp (the

additive _group

_{of p-adic}

integers).

As in the case of abstract free _{groups, any} closed

_subgroup

of a free pro-p-group is

again

free

((15, I-37,

Cor.

3]).

Furthermore,

let F be a free pro-p-group of finite rank and U be an _open

subgroup

of F. Then U is also

Fd-extensions. By

a

G-extension,

where G is a

profinite

_group, we mean a

Galois field extension whose Galois group is

isomorphic

to G as a

_topological

group. As a

generalization

of

Zp-extensions

([4]),

we consider

Fd-extensions,

and we introduce the

following

invariant: for a field k and a

_prime

_p, define

We shall write _p instead of

_pp(k~

if the reference to k and _p is clear from the context.

Examples.

(1)

If k is a finite

field,

then

pp(k)

= 1 for all _p.

(2)

If k is a local number

_field,

then we can

_give

an

_explicit

formula for

pp(k).

See

_Proposition

4.9 below.

Leopoldt

conjecture.

From now _on, let k be an

_algebraic

number field

(i.e.

a finite extension of the rational number field

Q).

Let us recall the

Leopoldt

conjecture

(cf.

[4, 2.3]).

Embed the _group of

_global

units E of k

diagonally

into the direct

_product

_Ilv,

where

_Uv

denotes the _group of local units of the

_{completion kv}

of k at v, and let E be the closure of the

image

of E with

_respect

to the natural

_topology

on

_Ilvlp

Uv.

Define the

non-negative integer

6p(k)

as

Then the

_Leopoldt

_conjecture

is stated as follows.

CONJECTURE 1.4

_(THE

LEOPOLDT

_CONJECTURE).

_{6p(k) =}

0 holds.

In other

_words,

_6p(k)

measures the

defect

of the

Leopoldt

_conjecture.

We

often write 6 instead of

_bp(k)

when there is no risk of confusion.

_Conjecture

1.4 is known to be true for all _p if k is an abelian extension

_{over Q}

or over an

imaginary quadratic

field

_{(Ax, Brumer).}

Several

_equivalent

formulations of

Conjecture

1.4 or

equivalent

definitions of 6 are known.

They

are of much interest in their _own, but the one that we need is the

following:

there exists

exactly

r2 -f 1 6

_independent

_{Zp-eztensions}

over

_k,

where _r2 denotes the

number

_{of complex places of}

k.

Let us return to

_{Fd-extensions.}

Since an

_Fd-extension

contains a extension as a

_{subextension,}

we have an

inequality:

In

_particular,

_p r2 + 1 holds if the

_Leopoldt

_conjecture

is valid for k and p.

Examples

with p

= _{r2 +} 1.

Example

1.6

_(1.

R.

Safarevic

_[14,

_§4]).

Let k =

Q(pp)

be the

p-th

cyclo-tomic field and assume that _{p is}

regular

(i.e.

_p does not divide the class

number of

_k).

Then the Galois _group of the maximal

_{pro-p-extension}

of k unramified outside _{p is} a free _pro-p-group of rank

(p

₊

1)/2

_{== T2 + 1.}

Since the

_Leopoldt

_conjecture

is true for k and _p, we

find, by

(1.5),

that Let

_GSp

denote the Galois _group of the maximal

_{pro-p-extension}

of k which is unramified outside _p. A necessary and sufficient condition on k

and _p for

_GSp

to be free _pro-p has been known

_(cf.,

for

_example,

_{[9, 2.1],}

[21,

Cor.]).

In this case, the rank of

Gsp

is

equal

to r2 + 1. Since the

Leopoldt conjecture

at _p is valid for such

_k,

we find that _p = _r2 + 1 holds

for such k and _p.

_{(Alternatively,}

use Lemma 2.1 instead of the

_validity

of

the

_Leopoldt

_conjecture

to

_{conclude p}

= _{r2 +}

1.)

A. Movahhedi and T.

_Nguyen

_Quang

Do

_{[9] (see}

also

_[8])

define such an

algebraic

number field k

_(i.e.

for which

_GSP

is free

_pro-p)

to be

_p-rational,

and

_{they investigate interesting}

arithmetic

_properties

of

_p-rational

fields.

Simultaneously,

G. Gras and J.-F. Jaulent

_[2]

introduced the notion

_of

p-regular

number field. This notion

_is,

in a _sense, a natural

_{generalization}

of

the

_regularity

of

_primes.

We refer the reader to

_[5]

_concerning

these

_topics.

Example

1.7

_{(cf. [8,}

_p.

_166]).

For a

prime

_p &#x3E;

_5,

an

imaginary quadratic

field k is

_{p-rational if p}

does not divide the class number of k. The converse

is not true in

_general.

Other

_examples

of

_p-rational

fields are found in

[8]

and

[5].

On the other

_{hand, by}

a result of K.

Wingberg

[18,

Kor.

3.3],

we can see the existence of non

p-rational

k

with p

= _{r2 + 1.}

Problems. A

_complete

determination of _p seems to be difficult. In this

note,

we shall focus on the

_following

two

_problems,

in view of the

_inequality

(1.5)

and known

_examples.

Problem 1.8. Prove

_p

r2 + 1 without

assuming

the

Leopoldt

conjecture.

As we mentioned in the

introduction,

we shall

_give

partial

answers to

these

_problems

in what follows.

2. Unramifiedness outside p of

Fd-extensions

The aim of this section is to _prove the

_following

LEMMA 2.1. An

_Fd-extension

over a finite

algebraic

number field is

un-ramified outside

_primes

above _p.

Remark. This is well known when d =

1,

([4, 2.2]).

Proof.

We shall

_give

two

_proofs.

First Proof: archimedean

_primes

do not

_ramify

in an

_Fd-extension

since

Fd

has no non-trivial torsion element. Let K be an

Fd-extension

over an

algebraic

number field

_k,

.~

_{(resp. 1)}

be a non-archimedean

prime

of K

(resp.

of

_k),

such that

_{C ~}

_p. Consider the localization

_Kzlkf.

This

is also a free

pro-p-extension

because

Gal(Kz/kt)

_may be considered as a

closed

_subgroup

of

_Gal(K/k).

But

_by

the

_assumption

_k(

_possesses a

unique

Zp-extension,

namely

the maximal unramified

_{pro-p-extension}

_{k r}

(cf. [4, 12.1]),

and

_consequently

is also the

_unique

non-trivial free

pro-p-extension

of

kt.

Therefore

_Ke

coincides either with

_kur

or with

_kj,

hence

is unramified over

k f.

0

Second Proof

_(due

to M.

_Asada):

as we mentioned

above,

the case d = 1

is well

_known,

which we admit in the

_following.

Therefore a

multiple

Zp-extension

_(i.e.

a

_Z~-extension

for some e

oo)

over a finite

_algebraic

num-ber field is unramified outside _p.

_Leth’/k

be as in the first

_proof

with its

Galois _group G ~

Fd.

Put

_Gi

=

G,

and

inductively

Gn+i

=

Gn]

(the

Frattini

_subgroup

of

_Gn).

Then the

_descending

series of closed sub-groups

has the

_following

_properties

_(cf.

_[15,

_1-38]):

(1)

is an _open normal

_subgroup

of G for all n &#x3E;

_1,

(2)

is abelian for all n &#x3E;

_1,

Let

_kn

be the fixed field of

_Gn.

_{Corresponding}

to

_(1)-(3),

the tower of fields

satisfies the

_following

_properties:

( 1 )’

is a finite normal extension for all n &#x3E;

1,

(2)’

is an abelian extension for all n &#x3E;

1,

00

(3)’

U kn

= .

n=1

Since is a

_quotient

of the maximal abelian

quotient

and since

_Gnb

is

_isomorphic

to

_Z~

for some e _oo,

kn+1/kn

is embeddable in a

multiple

Zp-extension,

hence is unramified outside _p.

By

virtue of

(3)’,

Klk

is unramified outside p. D

As is clear from the second

_proof,

any filtration of G

satisfying

(1)-(3)

will suffice. For

_example,

the

_descending

_p-central

series defined

_by

= The one that we

_used,

however,

has an additional

property:

(4)

is the maximal abelian

_quotient

of

_Gn

with

_exponent

_p, and this enables us to _prove the

_following

PROPOSITION

2.2. 1

Let k be an

algebraic

number field. Assume that the

Leopoldt

conjecture

with

_respect

_{to p} is true for _any finite

_p-extension

over k which is unramified outside p. If p = _{r2 +} 1

holds,

then k _possesses a

unique

Fp-extension

Klk,

and any

Fd-extension

(d p)

over k is contained

in 7~.

Proof.

Let 7~ be an

arbitrary

_Fp-extension

over k with

Galois

_group G.

Then,

with the same notation as in the second

proof

of Lemma

_2.1,

the

extension is characterized as the maximal abelian extension with

exponent

p which is embeddable in a

multiple

_{Zp-extension.}

Indeed,

the

maximality

can be seen as follows:

by

(4)

. Schreier’s formula

(1.1)

and note that the

_Leopoldt

_conjecture

for

kn

and _p is true. Thus K is

unique.

Let K’ be an

Fd-extension

(d

p)

and define

k~

in the same _way as we defined

kn

for K. Then we can

show k~

C kn

for all n &#x3E; 0. 0 3. Weak

_Leopoldt

_conjecture

In this section we recall the weak

Leopoldt

conjecture

and

apply

it to our

_problem.

Let k be an

algebraic

number

field,

k,,Ik

be a

Zp-extension,

and for each n &#x3E; 0 let

kn

be the n-th

_layer

of

_{k,,Ik, (i.e.}

_kn

is the

_unique

subfield such that

_{[kn : k]}

=

p~).

CONJECTURE 3.1

_(THE

WEAK LEOPOLDT

CONJECTURE).

bp(kn) is

bounded as n - oo.

Note that is a

non-decreasing

_sequence of

_non-negative

inte-gers.

For the

_importance

of

_Conjecture

3.1 in Iwasawa

_theory,

see

[3].

See also

_{[12, §2], [13, §3],}

or

[19, §5]

for Galois

cohomological

treatment. An

application

is found in

_[22].

We abbreviate "the

_Leopoldt

_conjecture

for k" to

_LC(k),

and "the weak

Leopoldt conjecture

for

_k~~&#x3E;"

to

_WLC(k~),

_(note

that whether

_Conjecture

3.1 is true or not

_{depends only}

on but not on the

ground

field

k).

We omit the reference _{to p} since it is fixed. The

_following

facts are known

concerning

WLC.

(3.2)

is true for the

_cyclotomic

_Zp-extension

(3.3)

If

_LC(k)

is

_true,

then is true for _any

_{Zp-extension kclk.}

(3.4)

Consider the set E of all

_{Zp-extensions}

of

_k,

and let ~’ denote the subset of E

_consisting

of those

_{Zp-extensions}

such that

is true. Then E’ is an _open dense subset of E with

_respect

to the natural

topology

on E

_([1,

Thm.

_{3], [12,}

Thm.

_2.11].

See also

_[3,

_§3]).

Our result in this section is the

_following

PROPOSITION 3.5.

_{If K/k}

is an

_Fd-extension

with the condition d &#x3E;

then is false for _any

_Zp-extension

contained-in

K. In

particular,

the

_following

assertions hold.

(1)

If

_{WLC(k~) is}

true for all

_k~/k,

_{then p}

r2 + 1.

(2)

If 6 &#x3E; 0

_(i.e.

the

_Leopoldt

_conjecture

is false for k and

_p),

then _p

Tz + 1

_{+ 6}

_{(strict inequality)}

holds.

Proof.

For any finite subfield L of

K/k,

is also a free

pro-p-extension,

and the rank is

_{given by}

Schreier’s formula

_(1.1):

The last

_equality

follows from unramifiedness at archimedean

_primes,

_(see

Lemma

_2.1).

Then it is clear that

_{(L : k~,}

and this _proves the first

statement.

₍₁₎

follows from this. For

₍₂₎

use

(3.2).

D

Question.

Can we

improve

this

by

using

(3.4) ?

4.

_Application

of a theorem of K.

Wingberg

In this section we

_compute

_p

_explicitly

in a

_special

case

(Corollary 4.6)

by using

K.

_Wingberg’s

"free

_product

_{decomposition"}

of Galois groups

(Theorem 4.5).

First we extend the definition

_(1.2)

of the _{invariant p} to a _pro-p-group

G as follows:

p(G)

:=

sup~d &#x3E;

_{0 ;}

G has a

quotient

isomorphic

to

_Fd}.

_(4.1)

(We

may allow d to be a cardinal

_number,

but in the

_following

G will

_always

be

_finitely

_generated.)

_Then,

for a field

k,

we have

where

_k(p)

denotes the maximal

_{pro-p-extension}

of k.

_Furthermore,

if k is

an

algebraic

number

field,

then

by

Lemma 2.1 we have

where

_{G sp}

is _{the Galois group of the maximal}

_{pro-p-extension}

of k which

is unramified outside _p

_(this

notation is consistent with that introduced

below).

Thus some information on

_Gs,

_may be useful in

_determining

_p. The pro-p-group

_Gsp

has been

_extensively

studied

_by

_many

_people,

and among their works we shall

_apply

a result of K.

_Wingberg

to our

_problem.

We fix the notation.

k is an

algebraic

number field

(we

assume _p is odd or k is

_totally

imaginary,

therefore an archimedean

_prime

does not

_ramify

in a

p-extension),

S 00

is the set of archimedean

_primes

of

_k,

SP

is the set of

_primes

of k above _p,

S is a finite set of

_primes

of k such that S D

So

U

_sP,

ks(p)

is the maximal

_{pro-p-extension}

of k unramified outside

_S,

GS

:=

Gal(ks(p)~k),

gv

:=

Gal(kv(p)/kv),

where kv

is the

completion

of k at a

_prime

_v, and

_kv(p)

is the maximal

_{pro-p-extension}

of

_kv,

up is the group of

p-th

roots of

_unity,

6(F)

= 1 or 0

according

as F D lip or

F 1J

_tip for a field F of

characteristic 0

_(this

is irrelevant to

_(1.3)),

So

is a maximal subset of

S B

S~

_satisfying

THEOREM 4.5

_(K.

_Wingberg

_{~20~, [21]).}

With the notation and

_assumption

as

_above,

if there exists

_So

such that =

0,

then

where * denotes free _pro-p

_product

_{(cf. ~10~)}

and .~’ is a free _pro-p-group

with

Remark. In addition to

_above,

K.

_Wingberg

_proves that if 0 for any

So,

then

Gs

is a _pro-p

duality

_group

(or

strict

Cohen-Macaulay

in the

terminology

of

_[15])

of dimension 2. We shall not treat this case in this note.

Our result is the

_following

COROLLARY 4.6. With the same notation as

above,

0 for some

So,

t h en

°

where _-, = 1 or 0

according

as

(kv :

Qp] is

odd or even.

Remark. The formula for p stated in the introduction is a

special

case of

this

_by

virtue of

_[21,

Lemma

_2].

Examples.

(1)

p =

2,

k =

Q(v/--i),

where i _&#x3E; 0 is _a rational

prime

such that 1 7

(mod 8).

In this case _p =

_1,

while

r2 + 1 = 2.

(2)

p =

3, k

=

~(~,

Vï5)

_or k =

~(~,

-26).

In this case _p =

2,

while

_r2+l=3.

In each case,

Sp

consists of two

elements,

say,

SP

=

(vo ,

Take S’ =

S,,,, U

Sp

and

_So

= The condition

-lv7sso

= 0 _was checked

by

V. M. Tsvetkov

[17]

in case

(1),

and

by

L. V. Kuz’min

[6],

V. M. Tsvetkov

[17]

respectively

in case

(2).

Proof

of Corollary

4.6.

LEMMA 4.7. Let

_{G1, ... ,}

Gm

be

_finitely

_generated

_{pro-p-groups.} Then

holds.

LEMMA 4.8

_(J.

Sonn

_[16],

cf.

_[11,

_p.

_102]).

Let G be a

(pro-p)

Demuskin group. Then

p(G)

= 2 (rk(G) -

E),

where - = 1 or 0

according

as

rk(G)

is

odd or even.

We shall _prove these lemmas in

_§5.

Recall the structure of the local Galois _group

_G,

_{(cf. [15, 11-5.6]):}

Combining

this with the

_{preceding lemmas,}

we can

_compute

_p. 0

Remark. If we assume that S = _U

Sp,

k _D

Ap and k is a Galois

(1) Sp

= So

= and

GS

is a free _{pro-p-group of} rank _{r2 +}

_1,

or

(2)

Sp

=

{vo, vl},

So

= and

GS =

gvo

=

9Vl

is _a Demugkin _group

of rank

_{r2 + 2}

_(so

that _p = 1 ₊

[r2 /2]

where

[x]

= the

_greatest

integer

not

exceeding

x).

This is the case with the

examples given

above.

Proof of

Remark. First note that

_So

consists of a

single

element

if k J

_J-Lp.

Assume

_V£

= 0. Then the formula for

rk(0)

in Theorem 4.5 becomes:

rk(0) =

(2 -

g)

+

_{(1/g -}

1)rl

+

_{(2/g - 1)r2,}

where g

=

~Sp

and _T1 denotes

the number of real

_places

of k. Since

_rk(.F)

must be

_{non-negative,}

we have

either

_{(1) g}

= 1 or

(2) g

= 2 and _T1 = 0. This

completes

the

proof.

D

From this

_point

of

_view,

we cannot

_expect

more

_{interesting phenomena}

as

long

as we restrict ourselves to the case k J _{J-Lp and}

_k/Q

is Galois

(S

=

S~

_U

Sp

is no restriction in the

light

of Lemma

2.1).

In

particular,

under these

_assumptions

_(including

_Vso

=

0),

we cannot find _any

_example

with _p = _{r2 +} 1 other than the case that

GS

itself is free.

Finally,

combining

(4.2),

Lemma 4.8 and the known structure of the

Ga-lois groups of the maximal

_{pro-p-extensions}

of local

_fields,

we can _compute

p for local fields.

PROPOSITION 4.9. Let k be a local

_field,

(in

the sense that it is obtained as

the

_completion

of an

algebraic

number field with

_respect

to a

non-archimed-ean

prime)

with residue characteristic t. Then

where - - 1 or 0

_according

as

[k :

Qp]

is odd or even. 0

Remark. For another

_proof

based on a

viewpoint

of

embedding

_problem,

see

[11, §5].

5. Proof of the lemmas

Proof

of

Lemma

_~.7.

_Firstly,

if there exist

_{surjective homomorphisms}

Fd,

(1 i

m),

then we can construct a

surjective

homo-morphism

- -

-m m

where d

_{(cf. [10,}

Satz

_1.2]).

Hence we have

*

j= i=i i=1

In order to obtain the inverse

_inequality,

we claim that if there exists a

surjective homomorphism

Gi ~

Fd,

then there exist a

partition

d =

i=1

m

(d2

&#x3E;

_0),

and

_surjective

_{homomorphisms pj :}

_Gi

-

(1 i

m).

i=1

We may assume m = 2. Let us consider each

Gi

as a closed

_subgroup

of

G, * G2 .

Then p induces a

homomorphism 1b :

Gi *

GZ -~

*

~p(G2 ).

On the other

_hand,

there exists a natural

(surjective)

_{homomorphism h :}

-~

_Fd,

and it is clear that _~p - h

o 0.

The

images

(i

=

1, 2)

_are free _{pro-p-groups,} and we have

since h is

_surjective.

_Starting

from the natural

_{surjections Gi ~}

(i

=

1, 2),

and

taking quotients

if _{necessary, we} obtain _a desired

partition

d

_d,

+

d2

and

_{surjective homomorphisms}

_{Fd, , (I}

=

1, 2).

0

Proof of

Lemma

_4.8.

Let G be a Demu0161kin _group. We must show that there

exists a

_surjective

_homomorphism

from G to

_Fd

if and

_only

if

_{d 2}

_rk(G).

This is a

_part

of a theorem of J. Sonn

[16,

Thm.

7],

but in the case

q(G) -

2,

he

_only

stated this and _gave no

proof.

We therefore _prove this

along

his

_idea,

for

_{completeness.}

Note that the "if"

_part

is obvious from the canonical form of the "Demuskin relation"

_{(cf. [7]).}

In the

_proof

of the

_"only

if"

_part,

the

_key

_point

is that Lemma 5.1 below holds even in

characteristic 2. Now let G -~

_Fd

be a

_surjective

_{homomorphism.}

Put

M = and N = where

Z /pZ

is acted

trivially.

Both M and N are

Z/pZ-vector

_spaces, and we

_identify

N with its

_image

in M under the inflation _map. Then the

_{following diagram}

is commutative:

Thus ~V is an

_{"isotropic" subspace}

of the

_"symplectic"

_space .M. Therefore

it suffices _{to prove} the

_following

LEMMA 5.1. Let t be a

field,

M be a finite dimensional _{vector space} over e

and N be an

isotropic subspace

of M with

_respect

to this

pairing

~i. e.

a~b = 0 for aI~ _a, b

_{E N).}

Then

_dim(M).

Proo f.

Put r =

dim(N).

Then, by

an induction on _r, we can choose

_linearly

independent

vectors

_{b1, ~ ~ ~ ,}

_ar,

_bT

E M such that

_ar)

is a basis of

_N,

and = 1 if i

== j,

0 otherwise. 0 D

Remark. If t is not of characteristic

_2,

we can

_impose

_0,

i.e. we can choose _a,,

b,}

which forms a

_part

of a

symplectic

basis of

M

_(cf.

_[16,

_Prop.

_3]).

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Masakazu _Yamagishi

Department of Mathematical _Sciences,

University of _Tokyo,