Maximal unramified extensions of imaginary quadratic number fields of small conductors

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

K

EN

Y

AMAMURA

Maximal unramified extensions of imaginary quadratic

number fields of small conductors

Journal de Théorie des Nombres de Bordeaux, tome

9, n

o

_{2 (1997),}

p. 405-448

<http://www.numdam.org/item?id=JTNB_1997__9_2_405_0>

© Université Bordeaux 1, 1997, tous droits réservés.

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

Maximal unramified extensions of

_imaginary

quadratic

number fields of small conductors

par KEN YAMAMURA

RÉSUMÉ. Nous déterminons la structure _{du groupe de Galois}

de l’extension maximale non ramifiée Kur de _chaque corps quadratique

imaginaire de

_{conducteur ~}

420

_(~

719 sous

_GRH).

Pour tous ces corps K,

l’extension Kur coïncide avec _K, ou avec le corps de classes de Hilbert de K,

ou avec le second corps de classes de Hilbert de K ou avec le troisième corps

de classes de Hilbert de K. Les bornes d’Odlyzko sur les discriminants et

les informations sur la structure des groupes de classes obtenues par l’action

du groupe de Galois sur les groupes de classes sont ici essentielles. Nous utilisons aussi des relations sur le nombre de classes et un ordinateur _pour

le calcul du nombre de classes de corps de bas degré pour obtenir le nombre

de classes de corps de degré plus élevé. Nous utilisons aussi des résultats

sur les tours de corps de classes, ainsi que notre connaissance des 2-groupes

d’ordre ~

26 et _{des groupes linéaires} sur des corps finis.

ABSTRACT. We determine the structures _{of the Galois groups}

_Gal(Kur/K)

of the maximal unramified extensions Kur of _{imaginary quadratic} number fields K of

_{conductors ~}

420

_(~

719 under the Generalized Riemann

Hy-pothesis).

For all such _K, Kur is _K, the Hilbert class field of K, the second Hilbert class field of _K, or the third Hilbert class field of K. The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is

es-sential. We also use class number relations and a _computer for calculation

of class numbers of fields of low _degrees in order to _get class numbers of fields of _{higher degrees.} Results on class field towers and the _knowledge of the _2-groups of orders ~ 26 and linear groups over finite fields are also used.

1. INTRODUCTION

Let K be an

_algebraic

number field

(of

finite

degree)

and its

max-imal unramified extension. Then the Galois group

Gal(Kur/ K)

can be

1991 Mathematics _{Subject Classification. Primary 11R32,11R11.}

Key words and phrases. Maximal unramified extension, imaginary quadratic number

field, discriminant _bounds, class field tower.

both finite and infinite and in

_general

it is

_quite

difficult to determine the

structure _{of this group. If .K has}

_sufficiently

small root

_{discriminant,}

then

Kur =

K,

that

_is,

K has no nontrivial unramified extension. This is the

case, for

example,

for the

_imaginary

quadratic

number fields with class number _one, the

_cyclotomic

number fields with class number _one, the real abelian number fields of

_prime

power

conductors ~

67

(see

[57,

Appen-dix]).

For some fields K with small root

_{discriminant,}

we can determine

Gal(Kur/ K).

The _purpose of this _paper is to determine the structure of of

_{imaginary quadratic}

number fields K of small conductors. For

_{imaginary quadratic}

number fields K of conductors ~ 420

_(~

719 under the Generalized Riemann

_Hypothesis

_(GRH))

we determine

and tabulate them for K with

_Kl,

where

_Kl

denotes the Hilbert class field of K.

_(If

_{Kur =}

_Kl,

then

_{Gal(Kur/ K) =}

_Cl(K),

the class _group of K

_by

class field

_theory.)

For all such

_K,

is one of

K, Kl, .K2,

or

_K3,

where

_K2

(resp. K3 )

is the second

(resp. third)

Hilbert

class field of .K. In other

_words,

coincides with the

_top

of the class field tower of K and the

_length

of the tower is at most three. If

_possible,

we

give

also

simple expressions

of

K1

and

K2.

Also for K = with

723 ~

d~

1000,

we tabulate

_except

for some d.

From now on let K =

Q(U2)

be an

_{imaginary quadratic}

number field

with discriminant d 0. J. Martinet stated in

_[34]

that if

_~d~

_250,

then

_{Kur = Kl}

_except

for 7

_fields,

_{for which he gave the} structure of

(We

note that

_H24

for K =

Q( 248)

in

[34]

is

_falser)

He also stated that this fact is

_{proved by using}

the methods which J.

_Masley

_{[35] (and}

later F. J. van der Linden

[50])

used for calculation of

class numbers of real abelian number fields of small conductors.

_They

used

_Odlyzko’s

discriminant bounds and information on the structure of class _groups obtained

_{by using}

the action _{of Galois groups} on class _groups.

In addition to their

_methods,

we use a

_computer

for calculation of class

numbers of fields of low

_degrees

_(we

use

KANT)

and then use class number

relations to

_get

class numbers of fields of

_{higher degrees}

_{(see §3).}

Results

on class field towers

_[5,

_{25, 29, 31,}

_49]

and the

_knowledge

of the

_2-groups

of

orders

26

_[17]

_{and linear groups} over finite fields

(see §4)

are also used.

We know that if 499

_(jd)

_{__}

2003 under

_GRH),

then the

_degree

K~

is finite

_(see

_§2).

For these

_d,

we want to determine

The

_key

fact is that _any unramified

_(finite)

extension L of K has the same

root discriminant as K:

_{rdL - ~}

_{rdK = M.}

_Thus,

if we

have

rdK

B(2N),

where

_B(2N)

denotes the lower bound for the root

1 The referee _pointed out that this was corrected _by Martinet in a _supplement to his

discriminants of the

_{totally imaginary}

number fields of

_{(finite) degrees &#x3E;}

2N,

then we

_get

K]

N. We do not know the real values of

_{B (2N)}

(except

for

N -

_4),

_however,

some lower bounds for

B(2N)

are known.

The best known unconditional lower bounds for

_B(2N)

can be found in

the tables due to F. Diaz _y Diaz

_[12].

If we assume the truth of

GRH,

much better lower bounds can be obtained. The best known conditional

(GRH)

lower bounds are found in the

unpublished

tables due to A. M.

Odlyzko

[38],

which are

copied

in Martinet’s

_expository

_paper

[34].

Let

Ki

be the

_top

of the class field tower of K: K =

Ko 9 K, 9 K2 C

...

is the Hilbert class field of

_Ki),

that

_is,

I is the smallest number with

=

Kl.

If we cannot

_get

_Ki]

_60,

which

_implies

=

Ki,

from available lower bounds for

_B(2N),

we need to

_judge

whether

Ki

has

an unramified nonsolvable Galois extension and this is

quite

difficult. For

the fields

_{Q( -423)}

and

_{G~( -723),}

we have

h(Ki)

=

1,

that

is,

I = 1 and we cannot

_get

_{~Kur : Kil}

60 from available lower bounds for

_B(2N)

(even

under GRH for

_Q(~/~723)).

420

_(idl

~ 719 under

_GRH),

we

_get

[Kur : Kl]

60 and our main

_problem

is to determine the

_degree

[Kl : Q].

In

_general,

it is difficult to determine

_{[K2 : Q],}

because it is very

hard to calculate the class number

_h(Ki)

of

_Kl.

_(Of

course, for K with

small

_Cl(K),

we can calculate

_{h(Kl )}

with the

_help

of a

computer.)

Now

let

_Kg

be the _genus field of

_K,

that

_is,

the maximal unramified abelian extension of K which is abelian over

_Q.

If d is the discriminant of K and

d =

did2 ... dt

is the factorization of d into the

product

of fundamental

prime discriminants,

then

_d2,

_{... ,}

_{dt ),}

and we have

which

_implies

As

_K9

is a

_{multi-quadratic}

number

field,

h(Kg)

can be calculated

_by

the

method in

_[51],

and we _may

_expect

that

[K2 :

(Kg)l]

is small for fields we

consider on the

_ground

of the

_following

_proposition

(§4,

_Proposition

_2).

PROPOSITION. Let L be the Hilbert class

_{field of}

the _genus

_field

_Kg

_of

an

irnaginary

abelian number

_field

K. Then

_for

_any

_prime

number _p with

p{

[L : Q],

the

_p-class

_group

Cl(P)

_(L)

_of

L is trivial or

_noncyclic.

For all K with

_idi

1000 such that

_h(Kg)

&#x3E;

_K],

which is

equivalent

to

_Kl,

we have

K2

= For _&#x3E;

(at

least)

two distinct

_prime

_{factors, however,}

for most

_K,

this

_inequality

holds. In

_fact,

if a

quadratic

_{subfield #}

K of

_Kg

has class number divisible

by

an odd

_prime

_p, then we have

_{ph(K)/[Kg :}

K~.

_Thus,

the

following

problem

arises

_naturally:

PROBLEM. Characterize the

_{imaginary quadratic}

number

_fields

K with

K2=(K9)1

The author

_expects

that this

_problem

can be settled

group-theoretically

and a similar

problem

can be

posed

for real

_quadratic

number fields. We

will discuss this

_problem

in the

_Appendix

2.

For 12 fields K with

_~d~

1000 for which we have verified

Kur ~

there exists an

S’4-extension

M of

_Q

such that

KgM/Kg

is an unramified

A4-extension,

where

_S4

_{(resp. A4)}

denotes the

_symmetric

_(resp.

alternat-ing)

group of

degree

four,

and therefore

Kur D

Except

for

_Q(V-856)

and

_{Q(B/2013996),}

we can charaterize such K

_simply

as

dE

d

for some

_quartic

number field E: If the discriminant of d of K is

divisi-ble

_by

the discriminant

_dE

of a

quartic

number field

E,

then

_Kg

has an

unramified

_{A4-extension.}

Then the normal closure of E is an

S’4-extension

of

_Q

unramified at all finite

_primes

over its

quadratic

subfield

Since

_dE

_I

_d,

is unramified

_by

genus

theory

and therefore

is an unramified

A4-extension.

Also for the fields

Q( -856)

and

_{Q(B/2013996),}

we can find

S4-extensions

of

Q

that

_give

their unramified

S4-extensions

by composition.

(For

details,

see

§7.)

Therefore,

data for

quartic

number fields are useful for our

_study.

_{Thus, K}

=

Q(J2)

with

Idl

1000,

can be classified

_simply

as follows:

Here, by

"dE

d" (resp.

d")

we mean "d is divisible

by

dE

for some

quartic

number field"

_(resp.

"d is never divisible

by

the discriminant

dE

for

any

quartic

number

_{field" ),}

and in the factorization of d =

d’dE, d’

denotes

a fundamental

_quadratic

discriminant. We

_expect

that

Kur = Kl

holds

for all I~ with

_Idl

_1507,

because we

_expect

that

Q ( -1507)

is the first

K

_having

an unramified nonsolvable Galois extension

(see

below).

This

for all fields .K with

_dE

_f

_{d 0}

_{-856, -996,}

= holds and that all

inequalities

for 1 are

equalties.

If our

_expectation

is

_true,

the classification

above can be

replaced by

the

following:

Though,

there are

possible

exceptions,

the author think that this

classifi-cation is

_meaningful

because this is

_complete

for

_Idi

_~

719 and even if an

exception

would occur for

Idl

&#x3E;

_719,

modification _{is easy.}

For most K we

_considered,

_{Kur = Kl}

is verified.

Thus,

the

following

natural

_question

arises: What is the first

_{imaginary quadratic}

number field

having

an unramified nonsolvable Galois extension ?

(What

is the first K

with

_Kur

_#

_Ki

_?)

Recent data for

_quintic

number fields

_{[3, 44]}

enable us

to

_give

a

partial

answer

(§8,

_Proposition

8):

PROPOSITION. The

_field

_{Q("; -1507)}

is the

_first

_imaginary

_quadratic

num-ber

_{field having}

an

unramified

A.5 -extension

which is normal over

Q

in the sense that none

of

of

discriminants d with 0 &#x3E; d &#x3E; -1507 has such

an extension.

We

_expect

that the field

_{Q( -1507)}

_gives

the answer to the

question

above.

For the determination of the structure of

_Gal(Kur/K),

the results on

2-class field towers due to H.

_Kisilevsky

_[25],

F.

_Lemmermeyer

_[29,

_31],

and E.

_Benjamin,

F.

_Lemmermeyer,

and C.

_Snyder

_[5]

are _very

helpful. They

give

us information on the structure of the Galois group

Gal(K 2 (2)

of

the second Hilbert 2-class field

K22~

of K over K _{in many} cases.

Now we

_explain

the notations in our table. In the

simple

expressions

of

Kl

and

_{ai,,3i and Ti}

denote any

algebraic

numbers

generating

the

ith cubic number field of

_signature

_{(1, ),}

the ith

_quartic

number field of

signature

(2, 1)

with Galois _group

isomorphic

to

_S4,

and the ith

_quintic

number field of

_signature

_{(1, 2)}

with Galois _group

_isomorphic

to

_D5,

re-spectively,

where we consider that the number fields of each

_signature

and

each

_type

_(of

_{Galois group of normal}

_closure)

are numbered _up to

conju-gacy

by

absolute values of discriminants.

(Here

we do not need to consider

nonisomorphic

fields with same

discriminants.)

G denotes the Galois _group

_{Gal(Kur/ K).}

As

_usual,

_Cn

is the

_cyclic

group of order n,

V4

is the four group, that

is,

V4 -

C2

=

C2

x

_C2,

_Dn

(n &#x3E; 3)

is the dihedral group of order

2n,

Q4n

(n &#x3E; 2)

is the

_generalized

quaternion

group of order

4n,

and

_,S’D8n

_{(n &#x3E;_ 2)}

is the semi-dihedral _group of order 8n:

(m ~

2, n &#x3E;

3)

denotes the _group of order 2mn

_{given by}

M2n

(n &#x3E;_ 4)

denotes the modular _group of order 2n

_{given by}

A4

is the double cover of

A4 : A4 ^--’

SL(2, 3).

For some

_2-groups

we use

_{designations given}

in the table

by

M. Hall and

J. K. Senior

_[17].

We note that T. W.

_Sag

and J. W.

_{Wamsley give}

minimal

presentations

for

_{all 2-groups}

of orders

_26[42].

For

_simplicity,

for some of

them we use the

_following

_designations

used in

[5].

rl t

denotes the _group

of order

2m+t+l

_{given by}

denotes the _group of order

_{given by}

We note that

_r~,2

=

641,3p,

r~,3

=

64rsC2,

r!,2

=

_64r3n2,

r~,3

64r 8 e.

The

_organization

of this _paper is as follows: In Section

2,

we review how

to obtain an _upper bound for

K]

by using

discriminant bounds. In

Section

_3,

we review some results on class number relations. In Section

4,

we _{describe the information} on the structure of class _groups that can

be obtained

_{by considering}

the action of Galois _groups on class _groups.

In Section

_5,

we review some results on class field towers. In Section

_6,

we describe how to determine for selected values of d. In Section

_7,

we describe fields

_having

an unramified extension not contained

in

_{(Kg) 1 -}

In Section

8,

we describe unramified nonsolvable Galois

exten-sions of

_{imaginary quadratic}

number fields. In the

_Appendix

_1,

we

explain

how to calculate class numbers of

_{S4-extensions}

of

_Q.

In the

_Appendix

_2,

we discuss the

problem

of

characterizing quadratic

number fields I~ with

KJ

Acknowledgements.

The author thanks Prof. R. Schoof for useful ad-vices

_[43].

He also thanks Dr. F.

_Lemmermeyer

for information on his

results

_[32].

Table of _{imaginary quadratic} number fields K = _{719 with} Kl

Continued

Supplements.

We

_expect

that all

_inequalities

for l in our table are

equal-ities. For this

_expectation

we

_give

the

following supplemental

data.

, , .

V4-extension

L which is an

S4

x

_C3-extension

of K. If

_L,

then

~K.~,T:L~=2,or4.

For K =

Q( -916).

D3

x

_C5-

Put L =

where denotes the maximal extension of

_{Q( V229)}

unrami-fied at all finite

_primes,

which is an

S4-extension

of

Q.

(Cf.

For K =

(~( -687),

Kur

= See

§7.

The index "w-ur" means

"weakly-unramified" . )

L is an unramified

_V4-extension

of

K2

which is an

S4

x

_C5-extension

of K. If

_L,

then

_{[fur : L]}

=

2,

or 4.

For K =

Q(ý-996). Gal(K2/K) c---

D3

x

_{C6. K2}

has an unramified

V4-extension L which is an

S4

x

_C6-extension

of K. We have

L] __

32

2. UPPER BOUND FOR

_K]

We review here how to obtain an _upper bound for

~K~

for a

_given

number field K

_{by using}

discriminant bounds.

In this section K denotes an

_algebraic

number field of finite

_degree.

If we denote

_by

_nK the

_degree

of

_K,

the

_nKth

root of the absolute value of the discriminant

_dK

of K is called root discriminant of K and denoted

_by

rdK:

The

_following

lemma is fundamental for our

_study.

LEMMA 1. Let

_L/K

be a

finite

extension

of algebraic

number

fields.

Then

rdL

=

rdK

if

and

only if

L/K

is

unramified

at all

finite

primes.

This can be

easily proved by

the transitive law of discriminants.

The

_following

estimates for rd =

rdL

_are known

(see

[39]):

Uncondition-ally

we have

as n - _oo, where C = 0.5772... denotes Euler’s

_constant,

and n = _nL

and _ri

_{(resp. r2)}

denotes the number of real

_{(resp. imaginary)}

_primes

of L. From

_this,

for all

_{imaginary quadratic}

number fields K = d 0 with

_499,

_{[K,,,, : K]}

oo. Under GRH we have

as n - oo. From

this,

for all K = 0 with

I dl

~

2003,

[Kur : K]

oo.

By

Lemma 1 we

immediately

obtain the

following

proposition,

which

tells us how to

_get

an _upper bound for

~Kur : K].

PROPOSITION 1. Let

_{B(n, rl, r2) (N 9 n}

= _{rl +}

2r2,

_{rl, r2}

nonnegative

integers)

be the lower bound

for

the root discriminants

_{of algebraic}

number

fields

F

_{of finite degrees}

_{(&#x3E;_ n)}

such that

_ri(F)/nL

=

ri /n

for

i =

1, 2,

where

_rl(F)

_{(resp. r2(F)) is}

the number

_of

real

_{(resp. imaginary)}

_primes

of

F.

_Suppose

that K has an

unramified

normal extension L

of degree

m.

Let H be a

_{positive integer.}

(i)

If rdK

then

_{[Kur : L]}

H and

therefore

h(L)

H. In

_{particular, if rdK}

_{B (2mnK, 2mrl (K), 2mr2(K)),}

then

_Kur

= L.

(ii)

If

h(L) -

1 and

_rdx

then

Kur

= L.

We note that the number 60 in

_(ii)

is the minimal order of the finite nonsolvable _groups, that is the order of

_A5,

the

_alternating

group of

degree

five.

Thus,

the

_knowledge

of

_good

lower bounds for

_{B (n, rl, r2 )}

_{is very}

im-portant

for our

_study.

The best known unconditional lower bounds for

B(n,

rl,

r2)

can be found in the tables due to F. Diaz _y Diaz

_[12].

If we assume

_GRH,

much better lower bounds can be obtained. The best known

conditional

_(GRH)

lower bounds are found in the

unpublished

tables due to

A. M.

_Odlyzko

_[38],

which are

copied

in Martinet’s

expository

_paper

[34].

All the

_{imaginary quadratic}

number fields K =

Q(~), -d

=

_{3, 4, 7, 8,11,}

19,43, 67,163

satisfy

the condition in

_(ii)

for L = K. In fact

by

the table

in

_[12]

we have

rdK

163

B(60 -

_{2, 0, 60}

1).

Therefore none of these

fields has _any nontrivial unramified extension.

Let K be an

_imaginary

quadratic

number field of small conductor. What

unramified normal extension can we take as L when we

apply Proposition

1 ? Since known lower bounds for

_{B (n,}

rl,

r2 )

grow

large

as n _grows

large

if

r2 /n

is

_fixed,

we need to take an extension L of

_large

_degree.

_By

the reason

described in the Introduction it seems that the best choice is L = the

Hilbert class field of the _genus field

_Kg

of K

_except

when we can

easily

_get

an unramified extension of

larger

degree.

In most cases we cannot conclude = L

only by

(i)

of

Proposition

1,

that

is,

we

_get

_only

L]

H

for some H &#x3E;_ 3. Therefore we need to show

_h(L)

= 1

_by

class number

calculation or

_using

criteria for class number

divisibility.

3. CLASS NUMBER RELATIONS

In this section we review some results on class number relations on

nor-mal extensions of

_algebraic

number fields. We will use them to calculate class numbers of fields of

_{large degrees}

from class numbers of subfields.

Let

_L/K

be a

(finite)

normal extension of

algebraic

number fields and G

its Galois group. For any

subgroup H

of

G,

we denote

_by

1~

the induced

character of G from the

_principal

character of H. S. Kuroda

_[27]

and R. Brauer

_[6]

_proved

_{independently}

the

_following

_{relation among the class}

numbers

_h(M),

the

_regulators

_R(M),

and the numbers

_w2(M)

of roots of

unity

of

_2-power

orders of intermediate fields M of

_L/K:

If we have a linear

relation _among

_{1 H}

then we have the

following

relation

Here,

LH

denotes the intermediate field of

_{L / K}

_{corresponding}

to H

_by

Galois

_theory.

We want to

_get

_h(L)

from the

_knowledge

of

_h(.K)

and

_h(M)

for

inter-mediate fields M. When an

_arbitrary

finite _group G is

_given,

there does

not exist

_necessarily

such a nontrivial

(that

is,

aH 54

0 for some

H)

lin-ear relation. Even if there exists a nontrivial linear

_relation,

it is diflicult

to use

(t)

in this form because it contains

_regulators.

However,

in some cases the

product

can be

simplified

into the form

[EL : E]/q

H

by elementary calculations,

where

_EL

is the _group of units in

_L,

E is its

subgroup

generated

by

E~

for some subfields M of

_L,

_{and q}

is some _power

of a

_prime,

and moreover

[EL : E]

is also some _power of the

_prime

_dividing

q. Such a

simplified

relation is often _very useful for calculation of class

numbers of fields of

_{high degrees.}

Let G =

( a,

b

I

a2n

= b2

=

_1,

b-lab

= be the dihedral

group of

order 2n

_{(n &#x3E; 2)}

_(the

four _group if n =

2).

Then we have the

_following

relation:

From this and some

_{elementary calculations,}

we

_get

the

_following

lemmas.

LEMMA 2.

_{(~27, 28))}

Let

_L/K

be a

V4-extension

of algebraic

number

fields

and

_F,

M and N its three intermediate

_fields.

Then we have the

following

class number relation:

where r is the Z-rank

_of

_EK,

t is the number

_{of infinite}

_primes

_ramified

in

L/K,

and v E

_{10, 11}

_(for

the

_{definition of}

v, see

[28]).

Moreover,

the index

The

_special

case K =

Q

and L D

Q( I)

of this is a classical result

due to

_Dirichlet,

which we can consider as a

_prototype.

_By

this lemma we can calculate class numbers of some fields of

_degree

24.

LEMMA 3.

_([18,

_19,

_36])

Let K be an

imaginary quadratic

number

field

or the rational number

field

Q,

and _p an odd

prime

number,. Let L be a

Dp-extension

o f

K. Let M and M’ be _any two intermediate

_{fields o f}

_L/K

with

_{[M :}

_K]

=

[M’ :

K]

= _p and N the

unique

quadratic

subextension

of

L/K.

Then we have the

f oldowzng

class number relation:

where b =1

_{i f}

K =

G~

and L is

imaginary,

and b = 2 otherwise.

Moreover,

the index

_{[EL : EMEM,EN]}

=

_pa

with 0 - a - b.

Furthermore,

if

L/N

is

unramifced,

then a =

b -1,

that

is,

Let K be an

_{imaginary quadratic}

number field whose

p-class

_{group is}

cyclic

of order _p. Then the Hilbert

_p-class

field of .K is a

_Dp-extension

of

Q.

If _p =

3,

or

_5,

_by

Lemma 3 we can

_compute

its class number

by calculating

the class number of its subfield of

_degree

_{p. For} this purpose, we use data

for cubic fields and

_quintic

fields. Data for number fields of

_{degrees n}

with

3 __

n _ 7 and small discriminant

_(in

absolute

_value)

are available

_by

anonymous

ftp

from

megrez.math.u-bordeaux.fr

(147.210.16.17).

LEMMA 4.

_([9])

Let L be an

_{imaginary D4-extension}

of Q.

Let M and N

be the two

_{nonisomorphic}

nonnormal

_quartic

_{sub fi elds}

and K its

_quadratic

subfield

such that

_L/K

is

_cyclic.

Therc we have the

f ollowing

class number

relation:

r ~ T""I T""I 1

Here, v

= 3

if

L is _a

CM-field

and _v = 2 otherwise.

Moreover,

the index

~EL : EMENEK]

is a

_nonnegative

_power

of

2.

If K is an

imaginary quadratic

number field with

cyclic

2-class _group of

order

4,

then

_by

this lemma the class number of the

_unique

unramified

cyclic quartic

extension L of K can be

computed

from the class numbers of

and all of its nonnormal

_quartic

subfields have odd class number. Therefore

we have

because the exact

_2-power

_dividing

_h(L)

is

h (2)

_(K)/4,

where

h~2~ (K)

is the 2-class number of K.

4. THE ACTION OF GALOIS GROUPS ON CLASS GROUPS

The action of Galois groups on class _groups can often be used to obtain useful information on the structure of class _groups.

We first review the

_following,

often called

_p-rank

theorem.

LEMMA 5.

_(See

_[53,

Theorem

_10.8])

Let

_L/K

be a

_{finite cyclic}

extension

of degree

n

of algebraic

number

fields.

Let _p be a

_prime

nurriber

with p f

n

and assume that all

fields

E with

K C E §

L have trivial

p-class

_group.

Then

_for

each

_{positive integer}

a, the

of

the

p-class

group

of

L is

a

multiple of

the

_{order f of}

_p modulo n. In

_{particlar, if}

_p

I

h(L),

then

pf

h(L),

and

_{if h(L)}

_pf,

The

_following

_{proposition gives}

us more information in a sense.

PROPOSITION 2. Let K be art

_imaginary

abelian numbers

field

and

Kg

its

genus

field,

that

is,

the maxirraal

unramified

abelian extension

of

K which is abelian over

Q.

Let L be the Hilbert class

field of

Kg.

Then

for

_any

prime

number _p with

_{p f}

_{[L : Q],}

the

_p-class

_group

Cl(p)(L)

_of

L is trivial

or

noncyclic.

Proof.

We first note that both L and its Hilbert

_p-class

field M are normal over

Q.

The Galois _group r =

Gal(L/Q)

acts

by conjugation

on

Gal(M/L),

which is

_isomorphic

to

Cl(p)(L)

_by

class field

_theory.

This action induces a

group

homomorphism

Now we assume that

CI~P~(L)

is

cyclic.

Then

Aut(CI~P~(L))

is abelian and

therefore the kernel of _p contains the derived _group of r. Hence if we let F

be the field

_{corresponding}

to

_Ker(p),

F is abelian over

_Q

and therefore F

is contained in

_Kg.

Since

_Ker(p)

=

Gal(L/F)

acts on

Gal(M/L)

_trivially,

by

the

_{assumption p f [L :}

_Q]

=

If

_we conclude that

Gal(M/F)

is factored

This

_implies

that the class number of F is divisible

_by

_{IGal(Mj L)I =}

and therefore

_I

_{h(Kg) .}

Hence

Cl(p)(L)

must be

trivial.

Remark. As described in the

Introduction,

if K is an

_{imaginary quadratic}

number

_field,

we can

_easily

calculate the class number of

_K9.

_Thus,

this

will be a

powerful

tool.

The _group

_homomorphism

p above often

gives

a useful information on

CI(l) (L)

also for

_prime

divisors l of

_{[L : Q].}

The same

_argument

works in

more

_general

situation. The referee

_pointed

out that this

_proposition

is a

special

case of a result due to O. Grfn with

_proofs

_(essentially

the same as

above)

by

L. Holzer and A. Scholz

_(Jaresber.

DMV 44

_(1934),

_74-75).

Let

_L/K

be a finite Galois extension of

_algebraic

number fields with

Galois _{group r. Let p} be a

prime

number and A the

p-class

_group of L.

The action of r on V = induces _{a group}

homomorphism

where r is the

_pl-rank

of A.

_Thus,

the

_knowledge

of the structure of linear

groups over finite fields is often useful for the

_study

of the structure of class

groups. Later we use the

following

facts.

LEMMA 6.

_([22,

_II,

Satz

_7.3a])

Let _p be a

_prime

and

_f

and n

_positive

integers.

GL(f, pn )

has a

_{cyclic subgroup}

S

_of

order

pin -

1.

(S

is called

Singer cycle.)

For _any

_subgroups

T

_of

S whose order is not a divisor

of

for

any e

f ,

its norrrialixer in

GL( f, pn) is

the semi-direct

_product

of

S with a

cyclic

_group C

of

order

f

such that the action

of

one

_generator

of

C on S is

_powering.

Remark. Let _q be a

prime

number

_p and

f

the order

of p

modulo _q.

Then _any

_{Sylow q-subgroup}

of

_{GL( f, p)}

is a

_subgroup

of a

_Singer

_cycle

and

its normalizer in

_{GL( f, p)}

has order

_{f (pf -}

_1).

LEMMA 7.

_([8])

Let _q be a _power

of

an odd

_prime

and W a

_{Sylow 2-subgroup}

of

GL(2, q).

If

q - 1

(mod 4),

then W is

isomorphic

to

_C2s

1

_C2,

the wreath

product of C2s by C2,

where 21 is the _{exact power}

_of

2

_dividing

_{q -}

_1,

while

if

q - 3

(mod 4),

then W is a semi-dihedral _group

of

order

2t+2,

zuhere

2t

is the _{exact power}

_of

2

_dividing

_q + 1.

Remark.

_By

this lemma

_{Sylow 2-subgroups}

of

_{GL(2, 3)}

and

_GL(2,5)

are

has three maximal

_{subgroups isomorphic}

to

_C4,

_D4

Y

_C4,

and

_SDls.

Here, D4

Y

_C4

denotes the central

_product

of

_D4

and

_C4.

Later for an unramified Galois extension L of an

_{imaginary quadratic}

number field which is normal over

_Q,

we consider the action of the

Ga-lois _group

_Gal(L/Q)

on the class group

Cl(L)

of L. Then an

_important

problem

is to determine the

_image

of the induced _group

_homomorphism

Aut(Cl(L)).

Thus,

we _prepare the

_following.

LEMMA 8. Let K be a

quadratic

number

field

and L its

finite

extension

which is normal over

Q. Suppose

that

L/K

is

unramified

at all

finite

Prirraes.

Then the Galois group

of

L/Q

is

generated by

elements

of

order

two not contained in the Galois _group

_of

_L/K.

In

_particular,

_Gal(L/Q)

is

isomorphic

to none

of

the

_{2-groups C4, Q8,}

SD16,

and

C4

I

C2.

The former assertion of this is a

_special

case of

[11,

_Corollary

16.31],

which is

_easily

deduced from

_{Cebotarev monodromy}

theorem. None of the

2-groups C4, Q8, SDls,

and

_{C4 62}

is

_{generated by}

elements of order two.

We note that if a

quadratic

number field has an unramified

quaternionic

ex-tension which is normal over

Q,

then its Galois _group over

Q

is

isomorphic

to

D4

Y

_C4

_([30]).

5. RESULTS ON CLASS FIELD TOWER

We also use results on class field tower. We review here some results.

The

_following

is well known.

LEMMA 9.

_{(See [49,}

Theorem

_I].)

Let K be an

algebraic

numbers

field of

finite degree and P

any

prime

number,.

If

the

p-class

group,

i.e.,

the

p-part

of

the class _group

_of

K is

_cyclic,

then the

_p-class

_group

_of

the Hilbert

p-class

_field

_{of K is}

trivial.

_{Moreover, if}

_p = 2 and the 2-class _group

of

K is

isomorphic

to

_V4,

then the 2-class _group

_of

the second Hilbert 2-class

_field,

that

_is,

the Hilbert 2-class

_{field of}

the Hilbert 2-class

_field,

_of

K is trivial.

This can be

easily proved by using

_group

theory.

All fields we consider have

_{cyclic p-class}

_group for _any odd

_prime

_p. In

fact,

a

noncyclic p-class

_{group for} an odd

prime

_p first occurs as

C1(Q(B/-3299)),

which is

_isomorphic

to

_Cg

x

C3

([7]).

On the other

_hand,

noncyclic

2-class groups often occur as the 2-class _group of

(a(f ), d

0

with small But for all fields we

_consider,

the 4-ranks of the class groups are at most one. The first

example

for

bigger

4-rank is d = -2379.

In the rest of this

_section,

we review some results on the 2-class field

towers of

_{imaginary quadratic}

number fields K:

K =

Ko2~ C Ki2~ c K22~ C ~ ~ ~

(Ki+i

is the Hilbert 2-class field of

K~2~.)

By

the lemma

_above,

_{if the 2-class group}

C1~2~ (K)

of K is

_cyclic,

then the

2-class field tower of K terminates with that

_is,

K22~

= When

does

K22~

= hold? F.

Lemmermeyer

[29]

gave a _necessary condition:

If

K22~

=

Ki2~,

then

CI(2) (K)

is

cyclic,

or

_isomorphic

to

_C2

x

_1).

He also obtained sufficient criteria in terms of the factorization of d. If

_{C1~2~ (K) ^_-’}

_V4,

then

K32~

=

K22~

by

the lemma above. We know that

any nonabelian finite groups of order 2’~ with n &#x3E; 2 whose abelianization is

isomorphic

to

_V4

is

_isomorphic

to

_{Q2n ,}

or

SD2,.. Thus,

if

(K) E£

Y4,

then

_{V4, D 2- Q2n,}

or

_SD2n,

where n is determined

by

2 n-2

₁₁

_{h(K (2))}

_(and

therefore n can be

easily calculated,

because

Ki2) _

Kg).

H.

_Kisilevsky

_[25]

characterized the occurrence of these groups in

terms of the factorization of the discriminant d of K.

When does

K32~

=

K22~

hold? The

complete

_answer has not been

given

yet.

However, Lemmermeyer

[29]

determined when

_{/ K)}

is

meta-cyclic.

(Of

course, if is

metacyclic,

K32

=

K2 )

by

Lemma

9.)

He

_proved

that if is nonabelian but

_metacyclic,

then

C1~2~ (K) ’-’

V4,

or d is of the form

_-4pq,

where _p

and q

are

_primes

congru-ent to 5 modulo 8:

(i)

If

_(plq)

=

1,

then

Gal(’ 2

/K)

is

isomorphic

to the group

given by

according

as the norm of the fundamental unit of the real

quadratic

number

field is -1 or 1.

_{Here, n}

is determined

_by

2n

₁₁

_h(Q(ýPq))

_(in

both

_cases),

and m is determined

by

11

In

[5],

these groups are denoted

by

MC,~

and

respectively.

We use these

notations in the

_Appendix

2.

(ii)

If

_{(p~q) _ -1,}

then

M2m+2,

where m is determined

_by

and that if we

_put

T = then

T2

= 1 and =

a 27n+l

.

Recently

E.

_Benjamin,

F.

_Lemmermeyer,

and C.

_Snyder

characterized K whose 2-class fields have

_cyclic

2-class _group

_[5].

_{They proved}

that

Gal(K 2 (2)

is

_{nonmetacyclic}

and its derived _group is

_cyclic,

if and

_only

if d is of one of the

following

forms: d =

-4Pp’,

where

p and

p’

are

_primes

with _{p -}

_1,

_{p’ -}

5

_(mod

₈₎

and

_(p/p’)

=

_1,

(p/p’)4(p’/p)4

=

_-1;

d =

-rpp’,

where -r is a

_negative

_{discriminant #}

_-4,

and _p and

_p’

are

_positive

prime

discriminants with

_{(plp’) = (r/p)}

= 1 and

(r/p’) _

=

-1.

_Moreover,

_{/ K)}

is

_isomorphic

to

_r:n,t

or

according

as

d =

-4pp’ (in

this _case

2t

=

h~2~(Q( -4p)),

_or

d

=

-rpp’

(in

this _case

2t

_{= h,~2~(Q( -rp)),}

and in both cases 2m =

h (2) (K)/2.

6. DETERMINATION OF

In this

_section,

we determine Our

procedure

for each K = is as follows:

(1)

We take a normal unramified extension L of K of

_{large degree}

for which we can obtain an _upper bound for

L]

less than 60

by

discriminant

bounds

_{(unconditional}

for

_idi

~ 420 and conditional for

_idl

&#x3E;

_420).

(2)

We show

_h(L)

= 1 and conclude = L

by Proposition

1.

(For

fields with

_cyclic

class _group of odd order

_prime

to

_15,

we can conclude =

K,

from

_{[Kur : Ki]}

168

_{(see below).)}

(3)

We determine the structure of

_(This

is not difficult for

most

_fields.)

As described in

_§2,

for most

_K,

we take L =

(Kg),.

The

_exceptional

cases are the

following

two cases:

(El)

d is divisible

_by

the discriminant

dE

of a

quartic

number field E and

the

_quotient

_d/dE

is a fundamental

quadratic

discriminant or 1. In this case

the normal closure M of E is an unramified

A4-extension

of the

quadratic

number field

_Q(J5)

and the

_composite

field KM is not contained in This case is divided into the

following

three subcases:

(a)

d =

dE

=

-p, where p is a

prime -

3

(mod 4):

d

= -283, -331, -491,

-563, -643,

-751. In this case, K has an unramified

A4-extension,

which

yields

a

quaternionic

extension of

Ki

=

(Kg),

by

composition.

(b)

d =

_dE

is _a

_composite:

d = -731 = 17.

(-43).

In this _case, K has

an unramified

_{A4-extension,}

which

_yields

a

V4-extension

of

Kl

=

by

composition.

d = -687 =

(-3) .

229, d

= -771 =

(-3) ~

257, d

= -916 =

(-4)’

229. In

this case, K has an unramified

S4-extension,

which

_yields

a

V4-extension

of

_(Kg),

_{by composition.}

(E2)

Though

K does not

_satisfy

the condition in

_(El),

we can check that

K has an unramified

S4-extension:

d =

-856,

-996.

These

_exceptional

cases will be treated in the next section. In the rest

of this

_section,

we treat the other fields. Let L = Then _we will show

K

We can treat fields with same class _group

_similarly.

_(Of

course, for some

fields,

additional consideration is

_needed.)

None of K with

_h(K)

= 1 has

any nontrivial unramified extension. For the fields K with

h(K)

=

2,

we

have determined in

_[58].

We first treat fields with odd class number &#x3E; 3. Note that

_Kg

= K if and

_only

if

_h(K)

is odd. We note that

_except

for fields listed in

(a)

no fields with odd class number _appear in our table as fields with

1 ~ 2. For all K with odd class number

_{and idl}

1000

_except

for d =

-787, -827, -859, -883, -947, -967, -971, -991,

we

_get

Ki]

60

by

discriminant bounds. Therefore our task is to prove

h(Ki)

= 1. We have the

_following.

PROPOSITION 3. Let K be an

_imaginary

_quadratic

number

field

with

_cyclic

class _group

_of

odd order n, &#x3E; 7. Assume that

_for

all intermediate

_fields

E

_{of K,IK,}

we have

El = Kl.

Then

if

h(Kl)

&#x3E; 1

_(this

is

_equivalent

to

K2 ~

Kl),

we have

26

= 64.

(This is

valid also

_for

real

quadratic

number

_fields.

In the case where K is

_imaginary

and

h(K)

is a

prime

(&#x3E;_

7),

64 can be

improved

to

132

= 169. See the

remark after

the

_proof.)

Proof.

Let 1 be a

_prime

factor of

h(K)

= n. Then

_by

the

_assumption

there exists a subextension E of

Kl/K

such that

h(E)

= l and

El

=

Kl.

Therefore

_{l ~ h(Kl)}

_by

Lemma 9.

Let p

be a

_prime

factor of

h(Kl)

and r the

p-rank

of the

_p-class

_group

of

_Kl.

We will show

_{p’’ &#x3E;__}

26

= 64. Since

by

Lemma 5 and

Proposition

2

we have

r &#x3E;

2 and 1

(mod n),

it suffices to eliminate the

following

four

_{possibilities:}

_(a)

_p

_{= 2, r}

= 3 if n =

7;

(b)

_p =

2, r

= 4 if n =

15;

(c)

p = 2, r = 5 if n = 31;

(d) p = 3, r = 3

if n = 13. Note that for all these

possibilities,

r is the order of _p mod n.

Put r =

Dn

and A =

Cn.

The action of r

If A n

_{1},

then we would have

I h(F)

for the field F

corres-ponding

to A n

_Ker(p),

which contradicts the

_assumption.

_(Cf.

the

_proof

of

_Proposition

_2.)

Therefore A n

_{Ker(p) _}

_{1}

and

_{p(A) =}

A ££

_Cn.

Since

Im(p)

is a factor group of F ££

_Dn,

_Dn.

Since A is a normal

subgroup

of

_r,

_Im(p)

is contained in the normalizer N of

_p(A)

in

_{GL(r, p) .}

Therefore N must have a

_subgroup

of

_isomorphic

to

_D~,.

Assume that n is a

prime

and that r is the order

_{of p}

mod n. Then

p(A)

is a

subgroup

of a

Singer

cycle

of

GL(r, p)

_satisfying

the condition of Lemma

6. Therefore N ^--’

_Cpr-l

x

_Cr.

Since N has a

_subgorup

of

_isomorphic

to

r must be even. This eliminates the

_{posiibilities}

(a),

(c),

and

(d).

We know

_GL(4,2)

=

PSL(4,2) ~

A8

[22,

II,

Satz

6.14,

(5)].

Since this

group does not have a

_{subgroup isomorphic}

to

_D15,

and

n = 15. This

completes

the

proof.

Remark.

_Regrettably

the same

_argument

does not work for

_h(K)

= 5: The

_{Singer cycles}

of

_{GL(4,2) ~}

_A8

have a

_{subgroup isomorphic}

to

_Ds.

However,

the class number relation

_{(Lemma 3)}

enables us to

_compute

_h(Kl)

by

calculating

class numbers of

_quintic

subfields of

Kl.

For all K with

h(K)

= 5

(all

such K have

discriminant ~

2683

[2, 52]),

_we have

h(Kl)

= 1

and therefore

_K2

=

Kl.

We _note that _{we can} find in

[20]

the class numbers of the unramified

_cyclic

_quintic

extensions F of K with 5

₁

_h(K)

and

~d~

1000. For all such

_K,

_h(F)

=

h(K)/5.

By

Lemma 3 we can check

that for all K with 3

_{1 h(K)}

and

_idl

1000

_except

for d listed in

_(a),

we have

_h(F)

=

h(K)/3,

where F is the

unique

unramified

cyclic

cubic extension of K.

F.

_Hajir

checked the

_parities

of

_h(Kl)

of K with

_Cq

_(q

an odd

prime

19)

and

_Idi

15000

_{by using elliptic}

units

_[16]:

For all such K

except

for d = -283

(q

=

3),

-331

(q

=

3),

-643

(q

=

3),

and -14947

(q

=

17), h(Kl)

is odd

(28

= 256

1 h(Kl)

for d =

-14947).

All K with

h(K)

= 7 have discriminant 5923

[2,

52]

and therefore

h(Ki)

is odd for

all such K.

_By

this reason in the case where

(K

is

_imaginary

and) h(K)

_{= q}

(q

an odd

prime &#x3E;_

7),

64 can be

improved

to

_{h(Kl) &#x3E;_}

132

= 169.

(Note

that

_GL(2,13)

has a

_{subgroup isomorphic}

to

_D7.)

_Moreover,

in the

case where

_{h,(K) &#x3E;_}

11 and

_21,63,

64 can be

_improved

to

h(Kl) &#x3E;_

2$ =

256

_(without

_assuming

the

_imaginarity

of

_{K.) (Note}

that

GL(8,2)

has a

_{subgroup isomorphic}

to

_D17.)

Suppose

Cn,

where n is an odd

_integer

_prime

to 15. The estimate 169 under the

_assumption

_h(Kl)

&#x3E; 1 is

_important.

_By

this,

if

_{[fur :}

_168,

then we can conclude =

Kl

under the

normal over

Q.

(Though

there exist

(probably

_infinitely)

_many

_imaginary

quadratic

number fields with

_cyclic

class _group of odd order

_prime

to 15

having

such an

extension,

the maximal discriminant of such fields is -2083.

(h(Q( -2083)

= 7. See

§8.))

For this

conclusion,

_we need

only

to show

that

_Kl

does not have an unramified

A5-extension.

(Note

that the second

minimal order of nonabelian

_simple

_{groups is}

_168.)

_Suppose

that

_Kl

has such an extension M. Then the Galois _group

Gal(M/K)

is an extension

of

_{A5 by}

_C~,.

Since any such extension is the direct

product

A5

x

_Cn,

K has an unramified

A5-extension

and this extension must be normal over

Q,

for otherwise its normal closure has

_degree

2 ~ 60z

= 7200. This contradicts the

_{assumption. Thus,}

_{[Kur : Kl~}

168

_{implies Kur}

=

Kl.

For K =

Q(~/Il827), Q( -859), Q( -967), Q(~/~991),

we do not

_get

Ki

60 but

_get

168

_by

_Odlyzko’s

_{(conditional)}

discriminant bounds and therefore =

Kl

(under GRH).

Now we treat fields K with even class number. For such

K,

K.

First we calculate

h(Kg ) .

We have two cases:

The

_equality

_h(Kg)

=

K]

is

equivalent

to =

Ki.

(This

holds

_trivially

in the case of odd class

number.)

If

Kur

=

Kl,

then

we have =

Kl.

We _expect the _converse, that

is,

if

(Kg),

=

Kl,

then

Kur = Kl

holds for fields considered. In

_fact,

we can show this for most

K.

We first treat fields with nontrivial

_cyclic

2-class _group. For such

_K,

its discriminant d is the

_product

of two

_prime

discriminants: d =

dld2

and

Kg

=

Q(,/d-,,

Since

h(K9)

but

_{h(K) I}

_h(Kg)

_by

Lemma

_9,

we have

h(Kg)

=

by

Lemma 2.

(Note

that

since

_d1

and

_d2

are

_{prime discriminants,}

both and are odd. More

_precisely,

we have

_ , F "r , , - - . F F , , r F .

where

_Cl(K)

x

Cl(K)

but

CI(K)2

= _E

CI(K)1,

be-cause the odd

_part

of the class group of a

biquadratic bicyclic

number

field is

_isomorphic

to the direct

_product

of those of its

_quadratic

subfields

(see

[28]).

Hence in this case =

h(K)/[Kg :

K]

is

equivalent

_to

h(Q( dl))

= = 1. We take

d2

as

d2

&#x3E; 0. How often does

= 1 occur? H. Cohen and H. W.

Lenstra,

Jr.

[10]

_posed

heuris-tic

_conjectures

_(the

so-called Cohen-Lenstra

_heuristics)

on the distribution

probability

that a real

_quadratic

number field with

_prime

discriminant has

class number one is about 0.75446 and this has been

_{numerically supported}

by

A. G.

Stephens

and H. C. Williams

_[48].

The first two discriminants of real

_quadratic

number fields with

_prime

discriminant and class

number &#x3E;

1

are 229 and 257. These are also discriminants of

_quartic

number fields! The

Cohen-Lenstra heuristics state also that the

_probability

that an

_imaginary

quadratic

number field has class _group whose odd

_part

is

_cyclic

is about 0.97757.

_Thus,

for

_simplicity,

we consider here fields with = 1

and

_Cn

_(n

an odd

integer).

Then

_h(Kg)

=

K]

is

_equivalent

to n = 1. When n =

_1,

under _some condition

_by

considering

Galois action as in

_Proposition

_3,

we

_get

an estimate for

h(Ki).

When ~ &#x3E;

_1,

we can

_easily

determine the structure of We first

give

an estimate for

h(Kl)

if n = 1 and then determine

Gal((Kg)l/K)

if

n&#x3E;1.

PROPOSITION 4. Let K be an

irraaginary quadratic

number

field

with

non-trivial

_cyclic

2-class group.

(i)

Assume

_C2m

_{(m &#x3E;_ 3)}

and that the

_unique

_{unrarraified cyclic}

quartic

extension E

_of

K has class number

2m-2.

_(The

latter

_assumption

irnplies

h(Kg)

=

h(K)/2.)

If

h(Kl)

_&#x3E;

1,

then we have

72

= 49.

(ii)

Assume

_h(K)

=

2q

(q

an odd

7)

and

h(Kg)

=

q

(=

h(K)/2).

If h(Kl)

&#x3E;

_1,

then we have

h(Kl) __&#x3E;

26

= 64.

(These

are valid also

for

real

quadratic

number

fields if

Kg

is the

unique

unramified quadratic

extension

_of

_K.)

Proof.

We

_{give only}

a sketch.

(i)

Put r =

Gal(Ki /Q).

Then r =

D2-.

By

Lemmas 7 and 8 any group

homomorphisms

F -

_GL(2,3)

and F -

_GL(2,5)

have

_image

of order at most

_eight

and this

_implies

that neither

_C~

nor

C5

can occur

_by

the

_assumption.

(Cf.

the

proof

of

Proposition

2. Note that

the

_{Sylow 2-subgroups}

of

_{GL(2, 7)}

are

isomorphic

to

SD32

and

they

have a

subgroup isomorphic

to

_D8.)

Hence we

_get

the conclusion.

(ii)

Put F =

Q( 04).

Then

D9.

Thus,

the same

_argument

as in the

_proof

of

_Proposition

3 works.

Remark. Note that when K has an unramified

cyclic quartic

extension

E,

we can calculate

h(E)

by using

Lemma 4. The fields with

Cl(K) ’-’

C2m

(m &#x3E;_ 2) and

1000 are divided into two

_types:

fields K with

h(E)

=

2 m-2

and fields K with =

2’n-lq

(q

an odd

prime).

For

most of the fields of the former

_type,

we can

easily

check =

Kl

by

using

the

_proposition

above. For the fields of the latter

_type,

we can check

Kur

= Then _we consider the Galois