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# On the ring of <span class="mathjax-formula">$p$</span>-integers of a cyclic <span class="mathjax-formula">$p$</span>-extension over a number field

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Texte intégral

(1)

a

over a

### number field

par HUMIO ICHIMURA

RÉSUMÉ. Soit p un nombre premier. On dit

extension finie,

### galoisienne, N/F

d’un corps de nombres F, à groupe de Galois G, admet une base normale

en

si

### O’N

est libre de rang un sur l’anneau de groupe

=

### O’F[1/p] désigne

l’anneau des p-entiers de F. Soit m =

### pe

une puissance de p et

une extension

de

m.

~

### F ,

nous donnons une condition nécessaire et suffisante pour que

F et

### p ~ [F(03B6m) : F],

nous montrons que

si et

seulement si

Enfin, si

p divise

### [F(03B6m): F],

nous montrons que la

### propriété

de descente n’est

vraie en

### général (Théorème 2).

ABSTRACT. Let p be a prime number. A finite Galois extension

### N/F

of a number field F with group G has a normal

basis

for

when

### O’N

is free of rank one over the group ring

Here,

### O’F = OF[1/p]

is the ring of p-integers

of F. Let m = pe be a power of p and

a

extension

of

m. When

~

### Fx,

we give a necessary and sufficient condition for

to have a

When

F

and p

we show that

has a

if and

if

has a p-NIB

divides

### [F(03B6m) : F],

we show that this descent property does not hold in

1. Introduction

We fix a

number p

let

be the

of

and

= the

### ring of p-integers

of

F. A finite Galois extension

### N/F

with group G has a normal

basis

for

when

### ON

is free of rank one over the group

It

has a normal

basis

for

when

### 0’ N

is free of rank one

over

For a

### cyclic p-extension N/F

unramified outside p, several results on

are

### given

in the lecture note of Greither

Let

### N/F

Manuscrit requ le 4 septembre 2003.

(2)

be such a

extension of

m =

In

it is known

that

E

it has a

if and

if N =

for

some unit E of

and

that

it has a

if and

if the

extension

has a

Theorem

is a fixed

m-th root of

### unity.

These results for the unramified case form a basis of the

of a normal

basis

for

### Zp-extensions

in Kersten and Michalicek

and

and

some

### corresponding

results for the ramified case.

Let m =

be a power

### of p, F a

number field with

E FX. In Section

we

### give

a necessary and sufficient condition

for a

Kummer extension

of

m to have a

It is

in terms

of a Kummer

of

but rather

### complicated compared

with the

unramified case. We also

an

### application

of this criterion.

When

FX and

we show the

descent

in Section 3.

Theorem 1. Let

be a power

a

### prime

numbers p, F a number

with

and K =

Assume that

a

extension

rri has a

and

has a

When p divides

this

of descent

does not hold in

we show the

### following

assertion in Section 4. Let

### Cl%

be the ideal class group of the Dedekind domain

### 0’ F

=

Theorem 2. Let F be a number

with

E F’ but

and

K =

### F((p2).

Assume that there exists a class C E

order p which

in

there exist

many

extensions

### N/F of degree p2

with N n K = F such that

has no

but

has

### a p-NIB.

At the end of Section

### 4,

we see that there are several

of p and F

the

### assumption

of Theorem 2.

Remark 1. In Theorem

the condition

means that

divides p - 1.

p must be an odd

as

### p t ~K : F].

Remark 2. As for the descent

of normal

bases in the

usual sense, the

### following

facts are known at

### present.

Let F be a number field with

and K =

For a

extension

of

### degree

p unramified at all finite

### prime divisors,

it has a NIB if and

if

### NK/K

has a NIB. This was first

Brinkhuis

### 

when p = 3 and F is an

and then

the author

for the

case.

When p =

for a tame

cubic extension

has a NIB if and

(3)

### only

if has a NIB. This was first

Greither

Theorem

### 2.2]

when p = 3 is unramified in

and then

the author

for the

### general

case.

2. A condition for

a

In

Theorem

Gómez

### Ayala

gave a necessary and sufficient condi- tion for a tame Kummer extension of

to have a NIB

the

usual

In

Theorem

we

it for a tame

Kummer

extension of

The

is a

version of these

results. Let m =

be a power of a

### prime

number p, and F a number field.

Let 2{ be an m-th power free

ideal of

for all

of We can

### uniquely

write

for some square free

ideals 31 of

to each other.

As in

### [4, 8],

we define the associated ideals

### j

of Ql as follows.

,m _ , 1

for a real

denotes the

~.

### By definition,

we have 0153o = 01531 =

### O’p,.

Theorem 3. Let m =

be a power

a

and F a number

with

E F.

Kummer extension

m

has

exist an E

with N =

such that

the

ideal is m-th

and

### (ii)

the

ideals associated to

are

The

### proof

of this theorem goes

to the

of

Theorem

we do not

its

the

of this

the conditions

and

in

Theorem

### 2]

are not necessary as m is a

unit of

### C~F-.)

It is easy to see that the assertion

### (A)

mentioned in Section 1 follows from this theorem. The

### following

is an immediate consequence of Theorem 3.

### Corollary

1. Let m and F be as in Theorem 3. Let a E

be an

such that the

ideal is square

the

extension

has a

Let

### HF

be the Hilbert class field of F. The

class field

of F

is

### by

definition the maximal intermediate field of

in which all

ideals of over p

Let

### ClF

be the ideal class group of

(4)

F in the usual sense, and P the

of

the classes

a

ideal over p.

we

have

class field

is

to

### Gal(HF/F).

It is known that any ideal of

in This is shown

to the classical

### principal

ideal theorem for

in Koch

pp.

### 103-104]. Now,

we can derive the

result

from Theorem 3.

### Corollary

2. Let m and F be as in Theorem 3.

### Then, for

any abelian extension

m, the

extension

has a

In

=

=

### 1,

any abelian extension

m has a

For

we write H =

For each

ideal £ of

we

can choose an

E

such that

=

the

### principal

ideal theorem mentioned above. Let E 1, ~ ~ ~ , Er be a system of fundamental units of

a

### generator

of the group of roots of

in H. Let

be an

### arbitrary

abelian extension of

m.

we

have

for some ai E

### Oi.

We see that NH is contained in

,~ runs over the

ideals of

al ~ ~ ~ as. As

is

the

ideal =

is square free.

the extensions

have a

### p-NIB.

As the ideal = square

the extension

ramified at the

### primes dividing

and unramified at other

ideals of

### 09. Therefore,

we see from the choice

### of (

and Ei

that the extensions in

are

### linearly independent

over H and that the ideal

### generated by

the relative discriminants of any two of them

the

has a

### p-NIB by

a classical theorem on

of

Frohlich and

III

### (2.13)]). Hence, NH/H

has a as NH C N. D

Remark 3. For the

of

### integers

in the usual sense, a result

to this

is obtained in

Theorem

### 1].

3. Proof of Theorem 1

The

if part follows

from

### [3, III, (2.13)].

Let us show the "if ’ part. Let m =

### pe, F, K

be as in Theorem 1.

(5)

p is an odd

number

Remark

Let

be a

extension

of

m, L =

and G =

=

Assume that

for some c,~ E

To prove that

has a

### p-NIB,

it

suffices to show that we can choose W E

such that

=

W.

### Actually,

when this is the case, we

see that

W. Let

=

=

and £ =

As

divides

p - 1

Remark

We fix a

m-th root of

Let

a be a fixed

of the

group

of

and let K E Z be

an

with

=

which is

### uniquely

determined modulo m. For an

x E

### Z,

let be the class in

=

x. For

1 ~

### f

e, the class in the

### multiplicative

group is of order f.

We

For each

e, we choose

### integers

r~i,’-’ E Z so that their

classes modulo

form a

set of

of the

we have

For

we put

a

of

### G,

we define the resolvents ao and of cv

for each

and

R.

### By (3),

we see that the

determinant of the m x m matrix of the coefficients of in the above m

is divisible

ideals of

p.

it is a

unit of

from the

=

cv, we obtain

Let

= and let

## C~L ~f’2’~ ~

x E

### 0’ L

such that xg = As

we see that

As is

### easily

seen, we have ao E

### 0’

and 0; f,i,j E From

W, we see that

K L L

(6)

the last

holds

from

we obtain

we

### Here, TrL~N

denotes the trace map. As we have

for 0 A m - 1. We see that the determinant of the m x m matrix of the coefficients of f,2,1 1 in the above m

is a unit of

I

we obtain

W .

as W E

has a

### p-NIB .

0

4. Proof of Theorem 2

Let

be as in Theorem

and

=

As

E

we can

choose a

of the

group

### AF

of order p so that

### (a2

p = p

with r, = 3 or 1 + p

### according

to whether p = 2 or

3.

we

2=0

The

### following

lemma is an exercise in Galois

### theory.

Lemma. Under the above

### setting,

let x be a nonzero elerrzent

K. We

Let L = Assume

is an abelian

extension

there exists a

extension

### N/F of degree

N r1.K = F and L = NK.

### Proof of

Theorem 2. Let C be as in Theorem

and 11 a

ideal of

contained in C.

the

of Theorem

is a

ideal.

= be an

ideal of

of

one in

to

and

is a

ideal of

### 0’ F splitting completely

in K. Let x = and define

an

### integer a by (7).

As = is invariant under the action of a, we

have

(7)

### according

to whether p = 2 or

3.

For

since r,’ -= 1 +

T - p mod

the last term

for some X E

We may as well

with

for

p = 2

it follows that

### according

to whether p = 2 or

3. In

we see

as p

in K

=

is a

ideal of

over p.

the

L =

is of

over

and there exists

a

extension

of

### degree p2

with = F and NK = L. We see

from

### (8)

and Theorem 3 that

has a

Let us show that

has

no

For

### this,

assume that it has

Let

### Nl

be the intermediate field of

of

p.

the

has a

We see

from

### (7)

and K - 1 mod p that =

with

As b E

and

### (p

E it follows that

for some 0 C

s p - 1. Since

= we have = As

has

a

### p-NIB,

it follows from Theorem 3 that there exists an

c E

with

=

such that

is

power free.

### Hence,

c =

for some I r p - 1

E F . We have =

As the

ideal

is

power

we must have =

This is a

### containing 0

is of order p. D We see in the below that there are many

of p and F

the

### assumption

of Theorem 2.

Let p = 2. Let ql , q2 be

### prime

numbers with ql - mod 4 and q2, and let F

the

field F

satisfies the

### assumption

of Theorem 2. The reason is as follows. Let 0 be the

### unique prime

ideal of over ql. We see that the class C = E

### ClF

is of order 2 from genus

Let K =

=

and

genus

### theory,

the class number of k in the usual sense

is odd.

we have for some

cx.

### Therefore,

= and the class C

in

### 0’ K*

Let us deal with the case

### p &#x3E;

3. Let p be an odd

### prime number, k

a real

field in which p remains

F =

and K =

Let

(8)

be the

extension of

### degree

p unramified outside p, and

=

we have K =

### FBI.

In the tables in Sumida and the author

we gave many

of p and k

### having

an ideal class C E

### Clk

which is of order p and

in

### (More precisely,

real

fields in the rows

= 0" and

### "no

= I" of the tables

the

For such a class

the lift

E

### ClF

to F is of order p and it

### capitulates

in K. As p remains

in

there is

one

ideal of

F

### (resp. K)

over p, and it is a

ideal.

we have

=

and

=

we obtain many

and F

the

of Theorem 2.

### Acknowledgements.

The author thanks the referee for valuable

which

the

### presentation

of the paper. The author

was

### partially supported by

Grant-in-Aid for Scientific Research

the

of

Science and

of

### Japan.

References

 J. BRINKHUIS, Normal integral bases and the Spiegelungssatz of Scholz. Acta Arith. 69

(1995), 1-9.

 V. FLECKINGER, T. NGUYEN-QUANG-DO, Bases normales, unités et conjecture faible de Leopoldt. Manus. Math. 71 (1991), 183-195.

 A. FRÖHLICH, M. J. TAYLOR, Algebraic Number Theory. Cambridge Univ. Press, Cambridge,

1991.

 E. J. GÓMEZ AYALA, Bases normales d’entiers dans les extensions de Kummer de degré premier. J. Théor. Nombres Bordeaux 6 (1994), 95-116.

 C. GREITHER, Cyclic Galois Extensions of Commutative Rings. Lect. Notes Math. 1534, Springer-Verlag, 1992.

 C. GREITHER, On normal integral bases in ray class fields over imaginary quadratic fields.

Acta Arith. 78 (1997), 315-329.

 H. ICHIMURA, On a theorem of Childs on normal bases of rings of integers. J. London Math.

Soc. (2) 68 (2003), 25-36: Addendum. ibid. 69 (2004), 303-305.

 H. ICHIMURA, On the ring of integers of a tame Kummer extension over a number field. J.

Pure Appl. Algebra 187 (2004), 169-182.

 H. ICHIMURA, Normal integral bases and ray class groups. Acta Arith. 114 (2004), 71-85.

 H. ICHIMURA, H. SUMIDA, On the Iwasawa invariants of certain real abelian fields. Tohoku

J. Math. 49 (1997), 203-215.

 H. ICHIMURA. H. SUMIDA, A note on integral bases of unramified cyclic extensions of prime degree, II. Manus. Math. 104 (2001), 201-210.

 I. KERSTEN, J. MICHALICEK, On Vandiver’s conjecture and Zp- extensions of Q(03B6p). J. Num-

ber Theory 32 (1989), 371-386.

 H. KOCH, Algebraic Number Theory. Springer, Berlin-Heidelberg-New York, 1997.

Humio ICHIMURA

Faculty of Science Ibaraki University

2-1-1, Bunkyo, Mito, Ibaraki, 310-8512 Japan hichimur~mx. ibaraki . ac . jp

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