On Galois structure of the integers in cyclic extensions of local number fields

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

G. G

RIFFITH

E

LDER

On Galois structure of the integers in cyclic

extensions of local number fields

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{1 (2002),}

p. 113-149

<http://www.numdam.org/item?id=JTNB_2002__14_1_113_0>

© Université Bordeaux 1, 2002, tous droits réservés.

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

On Galois

structure

of the

_integers

in

_cyclic

extensions of local number fields

par G. GRIFFITH ELDER

Dedicated to Professor Manohar L. Madan

RÉSUMÉ. Soit p un nombre

_premier,

K une extension finie du corps des nombres

p-adiques,

et

_L/K

une extension

_cyclique

ram-ifiée de

_degré

_pn.

On suppose que le

premier

nombre de ramifi-cation de

_L/K

satisfait b1 &#x3E;

_1/2.

_pe0/(p-1)

où e0 est l’indice de ramification absolu de K. Nous déterminons

_{explicitement}

la

structure de l’anneau des entiers de L comme

Zp[G]-module,

où

Zp

désigne

l’anneau des entiers

_p-adiques

et G =

Gal(L/K)

le

groupe de Galois de L.

ABSTRACT. _{Let p} be a rational

prime, K

be a finite extension

of the field of

_{p-adic numbers,}

and let

_L/K

be a

totally

ramified

cyclic

extension of

_degree

_pn.

Restrict the first ramification

num-ber

_{of L/K}

to about half of its

_possible

_values,

b1 &#x3E;

where e0 denotes the absolute ramification index of K. Under this

loose

_condition,

we

explicitly

determine the

Zp[G]-module

struc-ture of the

_ring

of

_integers

of

_L,

where

_Zp

denotes the

_p-adic

integers

and G denotes the Galois group

Gal(L/K).

In the

pro-cess of

_determining

this _structure, we

_study

various restrictions

on the ramification filtration and examine the trace _map

relation-ships

that result. Two of these restrictions are

_{generalizations}

of

almost maximal

_{ramification.}

Our method for

_determining

this

structure is constructive

_(also

_inductive).

We exhibit

_generators

for the

_ring

of

_integers

of L over the _group

_ring,

_Zp[G]

(actually

over where is the

_ring

of

_integers

in the maximal

un-ramified subfield of

_K).

_They

are determined in an essential _way

by

their valuation. Then we describe their relations. 1. Introduction

The Normal Basis Theorem says that in a finite Galois extension of

_fields,

L/K,

there is an element whose

conjugates provide

a field basis for L over K. As

_such,

this theorem

_explains

the Galois action on fields.

_Restricting

Manuscrit reçu le 2 f6vrier 2001.

This research was _{partially supported through} UCR grant no. _{99-28, University} of Nebraska

our attention to finite Galois extensions of number

_fields,

_L/K,

we _may ask about the Galois action on the

ring

of

integers D L

of L. This is tied to the arithmetic of the extension.

_By

a result of E. Noether

[10],

we know that in order for a normal

integrat

basis to exist it is necessary for the extension to be at most

_tamely

ramified.

For local number

_fields,

E. Noether’s result is not

_only

_necessary but also sufficient. To be

_precise

we fix a

_prime

_p, write

_Qp

for the field of

p-adic

numbers,

let K be _any finite extension of

_Qp,

and suppose that L is a finite

Galois extension of K with Galois _group, G. The

_ring

of

_integers

of L has

a normal basis over the

_ring

of

_integers

I~K

of

K,

if and

_only

if

L/K

is at most

_tamely

ramified. In other

_words,

there is an

integer

a E

_{D L}

whose

conjugates,

a E

_G},

_provide

a basis for

D Lover

_{D K, precisely}

when the ramification index of

_L/K

is not divisible

_by

_p

_{(i. e.}

_{’PK -}

_{D L}

_{= f 3L}

and

p f

e

where q3

refers to the

_{unique prime}

_ideal).

Outside of this

_situation,

when the ramification index is divisible

_by

p

(i. e.

the ramification is

wild),

our

_{understanding}

of the Galois mod-ule structure of

_~L

is

_extremely

limited. To

_gain

some control over the

variety

of issues that can _emerge, we restrict our

investigation

in this

pa-per to those extensions with

_cyclic

Galois group. So that we _may focus our attention

_completely

on the difficulties associated with wild

ramifica-tion,

we assume these extensions to be

_totally

wild. Therefore we

study

in this _paper the Galois module structure of the

_integers

in

_{totally ramified,}

cyclic, p-extensions.

Because

_L/K

is

_{totally ramified q3K -}

_~7L

_{= q3’}

where e =

[L : K~.

Because

L/K

is _a

p-extension

[L : K]

=

pn

for some n. Before we

explain things further,

we

adopt

some notation.

1.1. Notation.

~ Fix a

_{generator a}

of G =

Gal(L/K~.

~ Let

Ki

denote the fixed field of

Upi.

Relabel

Ko

:= K, Kn

:= L. ~ Let

Di

denote the

_ring

of

_integers

in

Ki

and

q3i

the maximal ideal of

0,.

~ _{Let 7ri} be a

_prime

element in

_{Ki, and vi}

the valuation of

_Ki

so that

Vi(1ri)

= 1.

~ _Let

_{bl b2}

...

_bn

denote the lower ramification numbers

_{of Kn /Ko}

[12,

Ch

_IV].

. For i

_{&#x3E;_ j &#x3E;}

0 let

_%;,;

denote the relative _trace from

Ki

_to

_Kj

and let

’

t À..

be the

_integer

such that

_q3;’

~ Let T be the maximal unramified extension of

_Qp

contained in

_Ko,

_eo

the absolute ramification index of and

_f

the

_degree

of inertia of

_{Ko /Qp,}

so that

[Ko : T]

= _eo,

[T :

Qp]

=

f .

~ Let

~ For i &#x3E; 1 let

_(D

be the

_cyclotomic

polynomial,

and define

_~1(x)

= x - 1.

~ Let

§(z)

be the Euler Phi function with

~(1)

:= 1.

~ Use to denote the floor function

_(the

_greatest

_integer

less than or

equal

to

_x),

while denotes the

_ceiling

function

_(the

least

_integer

greater

than or

_equal

to

_x).

Generalizing

[12,

V Lem

_4]

_(using

_[12,

III

_Prop

_7]

and

_[12,

IV

_Prop

_4])

one sees that for i &#x3E;

_j,

While Serre’s Book

_[12]

is our

primary

reference for local number

fields,

our main reference for

_integral

_{representation theory}

is Curtis and Reiner

~2J .

1.2. Structure of the

_Integers

over the

Group

Ring

Zp[G].

Moti-vated

_by

the classification theorems of

_{Integral Representation Theory,}

one is lead in a natural _way to ask for the

_Zp[G]-module

structure of

_Dn.

If Noether’s result is seen as a determination of

Dk[G]-structure

of

Dn

for

every subfield k of

Ko,

this is then one

_approach

to

_{generalization.}

In

_[11]

this

_approach

was

_{adopted by Rzedowski-Calder6n,}

Villa-Salvador and Madan.

_They

were motivated

by

Heller and Reiner’s finite classifica-tion of

_{indecomposable}

_{Zp[Cp2]-modules}

_[6],

and

_sought

the

_Zp[G]-module

structure of

_02.

Due to technical

_{considerations, they}

did not

_get

a

com-plete

result,

finding

it necessary to

_impose

three

_separate

restrictions on the first ramification number:

, ,

These

_{restrictions,}

in

_particular

the second and third restriction of

_(1.2),

have

_subsequently

_played

a role in

_{generalizations}

to n &#x3E; 2.

The first of these

_{generalizations,}

_[5],

was derived under what _may be viewed

_principally

as a

_{slight tightening}

of the third

_part

of

(1.2).

Under

- - - ,

the

_following

condition on the traces was determined:

This condition restricts the number of

_{indecomposable}

modules that _may

appear in to a finite set

_[5,

Thm

_4],

therefore

_enabling

the determination of the structure of as a

Zp[G]-module

[5,

Thm

5).

The other

_{generalization}

to ~, &#x3E; 2 is this paper. Here we tie the second restriction in

_(1.2),

which we call near maximale

rarrtification,

to a condition

on the traces that _may be used to

_greatly

_simplify

considerations in

_[11].

See

_§2.4.1.

We also introduce a new restriction on

ramification,

_strong

ramification,

tied

_similarly

to "trace" relations.

_Strong

_{ramification implies}

near maximal

_{ramification.}

_Together

_{they yield}

our main result:

Theorem 1.1. Let _p be _any

_prime,

_Ko

be any

finite

extension

of

the

p-adic numbers and

_Kn/Ko

be a

fully

rami fied

cyclic

extension

of degree p’~,

r~ &#x3E; 1. Assume that is

_strongly

_{ramified -}

that its

_{first ramification}

number

_{satis fies}

-If

G =

Gal(Kn/Ko)

and H denotes the

subgroup of

order

p, then the struc-ture

_of

the

_ring

_{of integers, i7n,}

as a is determined

_inductively

by:

These modules are

_defined

in

_Appendix

A. Other notation is

_defined

in

_§1.1.

Note that denotes the submodute

_of

M

_{fixed by H,}

while e denotes the removal

_of

a direct

summand,

so that

(M

e N

= M .

To

_put

_strong

_ramification

in some

_context,

note that the first lower ramification number

_bl

of a

cyclic p-extension

is either

peo/(p -1) (if

_Ko

contains a

_p-th

root of

_unity)

or an

_integer

which

peo/(p - 1)

with

_{gcd(bl, p) -}

1

_[12,

IV

_§2

Ex.

_3].

_strong

_ramification

restricts

_bl

to about one half of its

_possible

values.

1.3.

_Organization

of

_Paper.

In

_§2,

we will discuss the

_implications

of six different restrictions on the ramification

filtration, making

the connec-tion between these restricconnec-tions and trace relations our theme.

_Using

the

implications

of

_strong

_{rami fication,}

we

provide

Galois

_generators

for the

ring

of

_integers

in

_§3.

A

_{complete description}

of the Galois

_{relationships,}

at this

_point,

would determine the Galois module structure. This is

_given

in two

_steps.

In

_§4

we

recursively

reselect the Galois

_generators

_again.

This time we are careful to select them to be

_compatible

with other

pre-viously

selected Galois

_generators.

This

_gives

us the Galois

relationships,

but leaves open the

possibility

that

they

may be

entangled

with one an-other. In other

_words,

certain Galois

_generators

_{may appear} in more than one Galois

_{relationship.}

The second

_step

occurs in

§5

where we

_disentangle

these Galois

_{relationships.}

The section concludes with the structure

_{of D3 .}

The

_{dis-entanglement}

in

_§5

results however in more than the structure of

D3,

it determines the structure of the main result of this paper

implications,

and

_provide

some

_corollaries,

the first of which is the main result of

_[5].

The last

_section,

_§7,

contains directions for

_constructing

ex-amples

and some further remarks. We conclude the _paper with

_Appendix

A,

an

explanation

of our

Zp[Cpn ]-module

notation.

2. Ramification restrictions and trace relations

The

_study

of the Galois module structure of the

_ring

of

_integers

has been

_closely

tied to restrictions on ramification ever since E. Noether’s Normal Basis Theorem. Often the

_utility

of a restriction on the ramification numbers emerges

only

after the restriction is translated into a statement

about trace _maps

_(in

other

_words,

translated into a statement

_concerning

the existence of certain elements in the associated

_order).

For

_example,

tame

_{ramification,}

which restricts the lower ramification numbers below

0,

is

_equivalent

to the condition

_~7K

_(this

condition was

generalized

to other ideals

_by

Ullom

_[13]).

We

_begin

the section with a universal trace map

relationship,

a

relation-ship

that holds for all

_cyclic

_{p-extensions,}

whether ramified or not. We

emphasize

this

_fact,

since the other trace

_{relationships}

that we examine are all associated with restrictions on the ramification numbers. Next we dis-cuss stable

_{ramification,:}

If the first ramification number is

_large

_enough,

the other ramification numbers are determined. We also discuss almost imal

_{ramification,}

a condition that has

played

an

important

role in other

investigations

of the Galois structure of the

_integers

in

_wildly

ramified

_cyclic

extensions -

_notably

the work of Bertrandias

_[1]

_concerning

structure of the

integers

over the associated

_order,

and the work of

_Miyata

[9]

and Vostokov

[14]

concerning

the structure over

Do [G].

Note that "tame" and "almost maximal" are two extremes of ramification - one bounds the ramification numbers from above while the other bounds the ramification numbers from below.

_Following

our section on almost maximat

ramification

we

_interpret

the restrictions

_imposed

in

_[11]

as

_{generalizations}

of this condition. In the last

_part

of this section we will

study

_strong

ramification:

Since this restriction is

_stronger

than stable

_{ramification.}

and one of the more useful

_{generalizations}

of almost maximale

rami fication, namely

near maximad

_{ramification.,}

we use it and its _consequences to _prove our main result.

2.1. A Universal Trace Relation. The

_family

of trace map

relation-ships

described in this section hold in _any

_cyclic

_p-extension,

_regardless

of ramification. Besides their

_utility

in our

work, they

have additional

Definition: Define the realizable

_{indecomposables,}

to be the set of

indecomposable

_{Zp[GI-modules}

M,

for which there is an extension

L/K

with

_{Gal(L/K) ~}

G such that M appears as a

Zp[G]-direct

summand of

ÙL.

·

Question

A: Is

_SCpft

_equal

to the set of all

_{indecomposable}

modules ?

_(Because

of

_[11,

Thm

_1],

this

_question

is

_{only interesting}

for

n &#x3E; 1.)

The existence of these universal trace

_{relationships}

enables us to answer

to

_Question

A. The answer is "no" . But first we state the

_{relationships}

from which we _may draw this conclusion.

Lemma 2.1. The

_{following relationships}

hold irc _any

_cyclic

_{p-extenszorc:}

Proo f .

If

_{Kj / Kj-2}

is unramified or

_{only partially ramified,}

then one has

Trj-1,j-2.oj-1 = ~ j -2

and the result

_clearly

holds. If the extension is

fully

ramified,

then this is

_equivalent

to the statement that

_Ajj-1

This

_inequality

_may be verified as follows: If

pjeo / p - I )

then we have stable ramification so use

(2.2)

to find

_{that bj}

=

bj-l

+

pi-leo,

and

( 1.1 )

to evaluate the A’s.

_Otherwise,

as one _may

check,

from which the statement follows. For further

_details,

see

[3,

Lem

6].

0 We now turn our attention to

_Question

A and address it for G ^--’

_{Cpn .}

Note that because the answer for n = 2 is "no"

(see below),

the _answer is

"no" for all n &#x3E; 2.

2.1.1.

_Application:

The Case

_p2.

Based _upon Lemma

_2.1,

we can

strengt-hen the main result of

_[11].

Since

_Trl,o.ol,

we determine that _any

_{indecomposable}

summand M

of .02

must

_satisfy

C

(MuP),

where X^t denotes the submodule of X that is fixed

_{by y}

E G.

One may

easily

check that

(IZ2, Z; 1) (See

§A.3.

The notation is from

[2,

Thm

_34.32])

does not

_satisfy

this condition. As a _consequence,

_starting

with

_[11,

_p. 419

_(51)],

_setting

the

_exponent

of

_{(R2, Z; 1)}

_equal

to _zero, we

immediately

determine that:

Theorem 2.2.

_If

_{max{eo/(p -1),}

_{(peo -}

_p

_{+ l)/(2p - 1)},}

then

as

_{Zp[G]-modules,}

where these modules are described in

§A.3.

We conclude this section

_{by strengthening Question}

A.

This

_question

is revised and discussed further in

_§7.

2.2. Stable Ramification. In this section we consider the first restriction of

_(1.2).

_{Wyman proved}

that when a lower ramification number is

large

enough,

subsequent

ramification numbers are

uniquely

determined

[15]:

If the first ramification number is

_{large enough, namely}

_{eo / (p -1 ),}

one _says that the extension is

stably ramified.

Using

(2.2)

and

_(1.1),

one _may

verify

that under stable

ramification,

As a condition on the

_traces,

this _may be

_expressed

as

If the extension is not

_stably

ramified and

_Ko

contains a

p-th

root of

unity,

then the

_possible

second lower ramification

numbers, b2,

have been classified

by Wyman

[15,

Thm

_32].

A

_variety

of second ramification numbers are

possible,

therefore to

_{keep complications}

associated with the ramification filtration to a minimum it is

prudent

to assume stable ramification.

2.3. Maximal and Almost Maximal Ramification. If _any of the lower ramification numbers are divisible

by

_p, then

bi

= for each i

[12,

IV

_§2

Ex

_3],

and the extension is called

_maximalty

_ramified.

So

_long

as the extension is almost

_{maximally ramified, namely}

bl

+ 1 &#x3E;

_{peo / (p -1 ),}

we _may use

[12,

V

§3]

to find that

In

_doing

_{so, it} is

_helpful

to note that

_bi

is the ramification number of

[12,

IV

_§3].

_Therefore,

under this condition the

_idempotent

ele-ments of

_{Qp [G]}

_decompose

the

_ring

of

_integers:

where = 0. Each

Mi

may be viewed as a torsion-free mod-ule over the

principal

ideal

domain, R,

:= Torsion-free modules over

principal

ideal domains are free

[2, §4D]

and so

_by

_checking

Zp-ranks,

we find that

Note that under this severe restriction the

ring

of

integers

has a

particularly

Remarks 2.3. As a _consequence of this

discussion,

it

only

remains for us to

determine the structure of the

_ring

of

_integers

for

_bi

_{peo/(p 2013 1) -}

1.

Under this

_condition,

the ramification numbers are

_{relatively prime}

_{to p.}

2.4.

_Generalizing

"Almost Maximal" : Near Maximal Ramifica-tion. The trace

_relationship

which results from almost maximal

ramifi-cation can be rewritten as = Since _g it is

clearly always

the case that · So if

Trj,j-1(Dj) =F pDj-l,

it must be that

_PÙj-1;

in other

_words,

some of the elements in the

_image

of the trace have valuation which is too

small. . To

remedy this,

one may increase valuation

through

application

of

(Q -1).

This results in the

_following

_family

of restrictions:

The first case t = 0 is almost maximal

ramification.

The next case t =

_1,

that we call near maximal

ramification,

follows from stable ramification and

_{(peo -}

_{p +}

_{1)~~2P -}

_1).

Lemma 2.4. Let the extension be

_{stably rami,fied}

and let t &#x3E;

_0,

then

Remark 2.5.

_Compare

with

_[11,

Lem

_3].

Proof.

Note that the

_inclusion,

g follows from the

_inequality,

_Àj,j-1

+

_{tbi &#x3E;}

pi-1eo.

Because of

_(2.3)

we _may

replace

in this

_inequality

with

_{À1,0 -}

eo +

pi-leo,

and work instead with

Al,o

+

tbi &#x3E;

eo.

Clearly

À1,0

=

[((bi

₊

1)(P -l»/pJ ~

_{eo -}

~61

if and

only

if

((b1

+

_{1 ) (p -1 ) ) /p &#x3E;}

eo -

tbi .

One may

easily

check that this is

equivalent

to the

_hypothesis.

0

Instead of

_emphasizing

near maximal

ramification

as a

generalization

of almost maximal

_{ramification,}

we can

_emphasize

the

_analogy

with

[5].

In this

_section,

we do for the second condition in

(1.2)

what

[5]

did for the third condition. First in Lemma

_2.4,

the second condition in

_{( 1.2)}

is

_interpreted

as a relation among trace _maps. Then we

greatly

simplify

the

_argument

in

[1 1] ,

and _prove a much

_stronger

result -

already

stated as Theorem 2.2. Of

course, Theorem 2.2 was stated and _proven earlier in this _paper. The basis for that

_{argument however,}

is

_[11,

p. 419

(51)],

a statement that is derived near the end of

_[11].

Here we outline an alternative

approach

which can be used to

_{greatly simplify}

considerations

_throughout

_[11].

2.4.1.

_Application:

The Case

_p2.

One may

easily

check that the trace

relationship

associated with near maximal ramification restricts the number of modules that can _appear in

~2

from a list of

_4p

+ 1 to the

_following

list of nine:

_Z,

_E,

_’R,2,

_{(R2, Z; 1),}

_{(’R2, R1;}

_Àp-2),

_(R2,&#x26;;

_Àp-1),

_{(R2, Z EÐ}

a~-2 ),

(R2,

Z C

E;1,

ap-2 ) .

The notation comes from

[2,

Thm

34.32]

and is consistent with the notation of

_[11].

The

_advantage

of this restriction is clear: families of modules that arise as extensions

( e. g. (R2,

?Z1;

Ai),

i =

0,1, ...

,p -

2)

may contribute at most one member

(e.g. (~Z2,

a~-2)).

One may

verify

the

following:

if M is an

indecomposable

Zp[Cp2]-module,

then

where Q is a

_generator

of

_Cp2

and

H-1 (-, -)

denotes Tate

cohomology.

If one uses this

_property

_along

with the other

properties

listed in

[11,

Table

_2],

one can start with the nine modules listed above and _very

_easily

derive Theorem 2.2.

2.5.

_Generalizing

"almost maximal":

_rings

with the same trace. Another way to express the trace

relationship

associated with almost max-imal ramification is as

Trj,j-1(Dj) = Trj,j-1(Ûj-1).

In this section we

generalize

this

_{interpretation}

of the trace

_{relationship. Alternatively,}

one

may view this section as the

_development

of a

family

of trace map

condi-tions

_generalizing

_[5,

_Prop

_1,

p.

144],

or as a

generalization

of the third restriction in

_(1.2).

Lemma 2.6. Under Stable

_{Ramification, if}

_bk

+ 1 &#x3E;

pkfeo/(P -

1)1 -

pk,

then

_for

all m =

1,

2, 3, ...

Proof.

One may

easily

check that these conditions are

equivalent,

so we

need

_only

_{prove 1.} But first we note that

re-gardless

of

_{ramification,}

since

_Dm+k

C forces

₉

So the restriction on the ramification numbers is

only

to force

_Àm+k,m

Because of

_(2.3)

we need

_only

show that + eo.

Using

(1.1)

and

(2.2)

one can check that

Àk+1,0

=

keo

₊

bl

₊ 1 ₊

1)lpk+1 J

while,

ak,o

=

(k -

1)eo

₊

bl

₊

1 +

_1)/pkj.

Let j :=

pk ~eo/(p -1)~ -

(bk

₊

1),

then because of the

assumption

J

_p~‘.

_Therefore,

we need

only verify

that if _q =

1)1

and

_bk

+ 1 = J then

Let r :=

q(1n - 1) -

_eo

(so

0 r _{p -}

_1).

_Replace

_{-b~ -}

1 with

_-pkq

+ 8.

Replace -bk+i - 1

with

_{-Pkq-p keo}

+

6,

and eo with

q(p -1) -

r.

_Equation

Remark 2.7. To be able to talk about the "eventual behavior

_{of k,"}

consider a

fully

ramified

_{Zp-extension.}

_By

[15],

ramification in such an extension will

_eventually

stabilize - there will be a t &#x3E; 0 such that

_{bt &#x3E;}

Assuming

stable ramification

_{b1 2:}

_{eo/(p -1),}

note that as k grows the

assumption bk

+ 1 &#x3E;

p~

weakens. Once this

_assumption

holds,

note that condition 1. is

_equivalent

to

_(bl

+

_{+ eo / (p -1 ) (}

1-&#x3E;

_{eo/(p - 1).}

( 1-1 /p 1 )

&#x3E;

_{feo/(p - 1)1 - 1.}

This will

_clearly

be

true for

_{large enough}

k. Therefore this

_type

of condition is

_widespread.

In

any

fully

ramified

_{Zp-extension,}

there is a threshold value

_ko

such that for all k &#x3E;

_ko

and all m &#x3E; 1 Condition 2. holds.

2.6. Traces on

_Quotient

_Rings:

_Strong

_{Ramification. The action} of on

Ûi/Ûi-l

is _easy to describe

_{for j &#x3E;}

i. The trace acts via

multiplication by

p

for j

&#x3E;

_i,

and

_by

0

_{for j -}

z. In this section we extend this action

_{to j}

i. To do so, we

identify

with

(a)

and instead

_study

the effect _upon · The result of our

study

is the most

_important

"trace" relation in our

_catalog.

The ramification restriction associated with this

_relation,

_being

_stronger

that

_others,

will be called

_strong

_{ramification.}

From Remark

_2.3,

we _may assume that all ramification numbers are

relatively prime

to _p. Meanwhile from

_[12,

IV

_§3],

we know that the rami-fication number of

_{KjIKj_1’}

is

_bj.

Let a E

_Kn

and

_=1,

then

following

[12,

IV

_§2

Ex 3

_(a)],

we find that

for j

_n,

Let us

_{begin by}

_considering

the effect of on Choose _any a E

_i7n

such that

_bl

_{mod p.}

As one can

_easily

show

_{by repeated}

application

of

_(2.6)

=

vn((Q -

=

vn(a)

₊

(p -

1)bl

-0

_{mod p.}

Therefore may be

expressed

as a sum:

where p

E v E

~n

and

_vn(v).

Assume for the _purposes of this discussion

_{that v #}

_0,

so we _may assume that

1. Since = v mod

knowing

the valuation

of v means

_knowing

the effect of the Galois action _upon the valuation of

a E

_.onl.on-1.

We are

_{faced, therefore,}

with the

_following

_problem:

How

to determine the valuation of v?

To solve this

_problem,

we

_apply

_{(a - 1)}

to both sides of

_(2.7),

_yielding

Note that

_1)a)

=

vn (a)

₊

b2,

vn ((0’ - 1)J.t)

_&#x3E;

_+Pbl

and

=

+ bi.

If _{we assume}

that b2

(2p 2013 l)6i,

then

1)a)

vn((Q -

1)),

hence

_{vn((aP - 1)a)}

=

1)v),

and is

b2

&#x3E;

_{(2p - 1) bi}

and

_{vn (p) fl}

0

_{mod p}

so that

1) J.L)

=

+ pb1,

then &#x3E;

_{vn((Q -1)~),}

hence =

vn((Q -1)v),

and

vn(v)

is

_uniquely

determined to be If on the other

hand,

neither condition is

_satisfied,

we

_require

additional information

concerning

a to determine the valuation of v.

Clearly,

we should assume one of the

_{conditions, either b2}

(2p - 1)bl

or b2

&#x3E;

_{(2p - 1)bi.}

Under stable ramification these are

equivalent

to

_bl

&#x3E;

peo/(2p -

2)

and

_bl

_{Peo/(2p -}

₂₎

_{respectively.}

All other restrictions on ramification have been lower

_bounds,

so we choose

bl

&#x3E;

_2).

We will refer to this restriction as

_strong

ramification.

Note that

_strong

implies

both stable and near maximal ramification.

This discussion is

_generalized

in the

_following

Lemma:

Lemma 2.8. Assume

_{gcd(p, bl)}

=

_1,

peo/(2p - 2)

and k

_j

_n.

Given a E

_SJ~

and

_bi

_{mod p}

there exists a v with

Vj(v) =

+

_{(bk+1 -}

_{mod p,}

such that

_{4D p k (C)a =- v}

mod

_{D;-i .}

In

fact,

=

v+lt

where p E and =

Proof.

If

_b1

&#x3E;

_{1/2 - peo/(p - 1),}

then

_using

_(2.2)

_bk

&#x3E;

_{1/2 .}

pk eo/(p -

_{1) .}

(2 - pi-k) &#x3E;

1/2 ~

pk eo/(p -

1)

for k &#x3E; 1. Because

_bk

&#x3E;

_{1/2 ~}

pk eo/(p -

₁₎

and =

bk

₊

Pkeo

_we find that

bk+ 1

_&#x3E;

(2p -

If

v~ (a) -

bi

mod P,

then

_{v. (-(b pk (0’) a)}

=

vj (a)

₊

(p -1)bk ==

0

mod p.

As a _consequence, there exist

_{elements IA}

E

₇

such that

_{(D pk (o,) a}

= where =

3. Galois

_generators

As mentioned in the

_abstract,

we shall determine the

Zp[G]-module

struc-ture of

_{Dn by}

_exhibiting

_generators

and

_describing

their relations. In this section we determine the

_generators.

The

principal

mechanism

whereby

we

may conclude that a set

_generates

is the

_following

basic observation. Remark 3.1. Let

{3i

E

_Dj

be elements

_subject

_only

to the condition =

i. Then the set _{1 Iz=0} is a basis

_{for Oy}

over

DT-As a _consequence of this

remark,

we will work

principally

over

DT,

the

ring

of

_integers

in the maximal unramified subfield of

_{Ko .}

The most

im-portant

part

of our

work, especially

in this

section,

is our

development

of

.oT[G]-generators

for each

_quotient,

_{l, ... ,}

n. The

3.1.

_Quotient

module

_generators.

This section is devoted to the

de-velopment

of

_{DT[G]-generators}

for

_Ùj/Ùj-1.

Since

_{(0’) .}

=

0,

we _may follow the discussion in

§2.3, observing

that the

Or[G]-structure

of

_Ùj/Ùj-1

is

_really

In other

_words,

_Ùj/Ùj-1

may be viewed as a module over

sJT(G’~/(~~,; (Q)) =’

where

_~T,;

denotes a

_primitive

pj-th

root of

_unity.

Now

_I

is a

principal

ideal domain and modules over

principal

ideal domains are

_free,

therefore

for some

_exponent

_a, as where 0’ acts upon

3DT[Cpjl

via

multiplication by

_(pi.

To determine the

_exponent,

_compare

_OT-ranks.

One finds that a = _eo. So it is

appropriate

_to talk about a basis for

over Since

DT[lQ-structure

is

_essentially

.oT[(pi]-structure

we find it convenient to blur the

_distinction,

and refer to a as an

This should not cause too much confusion.

We will use stable ramification in this section and because of Remark 2.3 assume that the ramification numbers are all

relatively prime

to _p. Our main tools are

_equation

(2.6),

and the _consequence of

_strong

ramification,

Lemma 2.8.

3.1.1. A n

_{for Dj -}

_{Let am~ j} E

_Kj

with =

bl

_{+ pm.}

We

leave it to the reader to check the valuations and

_verify

that based _upon Remark 3.1 and

_(2.6),

the

_following

elements

_provide

an

_,T-basis

for

_Dj:

( 1-1)pam,j, ... ,

for 0

_{v3 eo}

+

_bl

_{+ pm} and

p~eo

+

_b1

+ + pm 7 and for each value of t =

1, 2, ... p

-3.1.2. An

_for

_Implicit

in the

_DT-basis

that we

just

listed is the

DT[O’pi-l]-structure

of

_.oj.

See

_[3,

p

631-633]

for further

details. We

_glean

from these elements the

_following

for

£)j/Dj-1:

So far we have

only

used stable ramification and the fact that the ramifi-cation numbers are

_relatively

_prime

to _p. In the

_following

section we will use the

_assumption

of

_strong

ramification.

3.1.3. An

_for

_Dj/Ùj-1.

The elements in

_(3.1)

_certainly

span

over since

_they

span over a

sub-ring.

We need

to _{discern among them now,}

_selecting

out

_only

those elements which are

necessary. The

following

lemma is our

principal

mechanism for

doing

so. Lemma 3.2. Assume that

_{1/2 ~ peo/(P - 1)}

_bl

_{peo/(p -}

₁₎

_{and j &#x3E;}

1.

Then

_for

each a E

_Kj

where

_{vj (a)}

=

bl

_{+ pm,} there is _an element _{v E}

Kj

with vj (v)

+ tbj

+Pm = tbl mod p,

and an

_{element p}

E with

Proof.

This result is _proven

_by

_repeated

_application

of Lemma 2.8. Let a E

_Kj

with

_{vj (a)}

=

_bl

mod _p, and let 1 _{r, s}

_j.

Then

_by

Lemma

_2.8,

=

vi where

Vj(Vl)

=

vj (a)

+

_{br )}

and =

(peo - (p -

1)bl).

Using

Lemma 2.8

_again

we find that = _{v* + uc*}

where = and = ₊

(bs+1- bs ) -

(peo

-p

One may

easily

verify

that under stable ramification

bs

p(p - 1)b,,

therefore

_{by renaming}

_p2 := _{1-’* +} and _{v2 = V*} we find that

have our result when t &#x3E; 1. 0

Later we will need a

partial

converse to Lemma 3.2. We state and _prove it now.

Lemma 3.3. Assume that

_{1/2 Peo/(p -}

₁₎

bl

Peo/(p -

1) and j

&#x3E; 1.

Given _tt E with

_{= bk}

+ then there are elements _{a, v E}

_Kj

such that =

bl

_{+ pm,}

=

_{+ pm}

₊

bj

and

Proof.

First we extend the above Lemma and _prove that

_given

a E

_Kj

with

_vj(a)

=

bl

_{+ pm}

there _are

_{elements p}

_E

Kj-l

and v E

_Kj

with valuations as

above,

such that

(QF’’ ’ - 1)p/(Q - 1) ~ a = v + ~.

Note that

(~~’-1 _ l~n~~~ _ 1~ -

_{(~’’-l)P-~-i(7)...$p(7).}

Using

the Binomial Theorem to

_expand

a# =

((oPj-l -

₁₎

+

_1)P

then

_{dividing by}

_{(Q - 1),}

we

find that

Lemma 3.2 we find that =

ito + vo where

Vj(/Jo)

=

(Peo -

and

_{= pieo}

_{+ pm + bj .}

Furthermore

(cr) - "

= vt where =

pieo

_+pm

_&#x3E;

+ Now it is easy to show

_{by using}

_[12,

V

_§3

Lemma

_4]

that

and v = and we have the desired conclusion.

We are

ready

to prove this lemma. To prove this we start _{with ao E}

_Kj

such that

_{vj (ao )}

=

vj (p) -

j#eo

+ bj - (peo - (p -1 ) bl ) )

+

_bl,

and elements

ai E

_K~

i =

1, 2, ... ,

such that =

vj(ao)

+ pi.

Then ==

b1

mod p

for i =

0, l, ... ,

and

using

Lemma 3.2 we find that there are

elements

E E

_Kj

such that

(Q~’ 1 -1 )~ / (~ -1 ) ~

_{ai = vz} + pj,

Vj-1(p,)

+ i. So there must be

units ui

E

,l:7o

such

_{that p}

=

I::o

uipi- Let a =

I::o

_uzaz and v =

I::o

_uivi. The lemma follows. D

As a _consequence of Lemma 3.2 we _{may prove} the

following

which is the main result of this section.

Lemma 3.4. Assume that

_{1/2 ~ peo/(P - 1)}

_bl

_{peo/(p - 1)}

_{and j &#x3E;}

1.

Let

Then the union

_of

these

_sets,

_namely

_By

:=

Bj,t, is

an

for

Furthermore

_{if (3}

E then

_{v~ (,~)}

=

(t

+

_l)6i

_{mod P}

and

Proof.

From our earlier

_discussion,

we know to look _among the elements of

Note that _any element of

_{3

E B such that

can not be in this basis. We combine this observation and Lemma 3.2 now.

Considering

those elements in B of the form we

easily

eliminate all elements

_except

those in Now we consider those elements of the form

E B.

_Using

the Binomial Theorem as we did in the

proof

of Lemma

_3.3,

we find that

So to determine the effect of on

E it is clear that we need

_only

determine

1)t-lp;- ()

E

_{Dj/.oj_1.}

Because of this and Lemma 3.2 we eliminate all elements E

_B,

_except

for those listed in

The Lemma now

follows,

as it is

_easily

shown that the elements which

remain,

namely

_{Bj ,}

continue to _span

_.ojlDj-1

over

furthermore,

exactly

eo elements remain. 0 3.2. Galois

_generators

for the

_ring

of

_integers.

This section summa-rizes our

_{accomplishments}

thus far. For each 0 m eo - 1 choose an a E with

_vo(a)

= m. Let the set of such a be called

Bo.

Proposition

3.5. Assume that

_{1/2 ~ peo/(P - 1)}

_bl

_{Pep/(p -}

_1),

and that the sets

_Bj

_{for j}

=

1, 2, ... ,

_{n are}

defined

_as in Lemma 3.4. Then

B =

_{generates Dn}

_over

DT[G] :

Proof.

This is clear. 0

4. Galois

_{relationships}

From the

_previous

section we inherit

DT[G]-generators

for

Dn.

These

generators

are not

_yet,

_{however, completely}

suited to our needs. When

selecting

these

_elements,

we did not consider Galois

_{inter-relationships.}

In

fact,

they

were selected based _upon their valuation alone. To

_remedy

this

fact,

in this section we choose certain of these elements

_again,

this time

paying

attention to Galois

relationships.

When we are

_done,

we will not

only

have

_{DT [G]-generators}

for

_Ùn,

but also have their

_{Or[G]-relationships.}

4.1. Galois

_{relationships:}

outline of

_approach.

We

_proceed

induc-tively,

first

_assuming

that the Galois

_{relationships}

_{among the elements of}

Dn-1 are

known,

then

_determining

the Galois

_{relationships}

_among the

and then

_{disentangling}

these Galois

relationships,

we discuss the _process

first.

4.1.1. Form

_of

Galois relation,. Consider the canonical short-exact

se-quence

where i refers to the

_injection

and 7r refers to the natural

_projection.

Since

is free over the elements of

namely

for a E

_{Dn, satisfy}

the

_equation

= 0. Meanwhile since

the

_generators

of a E are also a basis

they satisfy only

this one

_equation.

Now

4bp. (a) a

E

.on-1

and

_assuming

that we can _{express any} element of in terms of the

_generators

of the Galois

relation-ships

among the elements of

.on

which are not

_already

in will all be

expressions

of the form

Since = a

_change

of a

_generator

a

by

an element of

Dn-1

will

_change

the

_{right-hand-side}

of

_(4.1)

_by

an element of

pDn-1

and so these

_expressions

are

_only

significant

modulo

pDn-1.

However if for each a E

_{Bn ,}

the

_{right-hand-side}

of this

_expression

was determined modulo

PÛn-1,

the Galois module structure of

Dn

would be

_completely

determined

by

induction.

4.1.2. Valuation and the Galois relations. In this section we assume that the ramification numbers are

relatively

prime

to p and describe a _process

whereby

the

_{right-hand-sides}

of the

_expressions

in

_(4.1)

_may be deter-mined.

Based upon their

expression

in Lemma

3.4,

it is clear that all elements

of

_Bn

besides those in

_Bn,o

map to zero under the

_trace;

thus

_resulting

in trivial

_{right-hand-sides.}

This leaves us to determine the

_{right-hand-side}

of

_(4.1)

for each a E

_Bn,o.

But if for each i 1 we had

Galois

_expressions

for

elements vi

E with

_{= i,}

then we could

use the

following

result,

Lemma

4.1,

to choose the elements of

_again.

In other

_words,

we can first determine the

right-hand-sides

for

(4.1)

and then find a to

_provide

the left-hand-side of the

_equation.

Lemma 4.1. Given _any element v E

_Kj-l

with

valuations,

= i,

there is an elements a E

_Kj

with

_valuation,

_{vj (a)}

such that = v.

Proof.

The ramification number of is Use

_this,

the fact that the trace _maps fractional ideals of

_Dj

to fractional ideals of

_{D j-1}

_[12,

V

_§3

Based upon this

observation,

we are left to find

_expressions,

in terms

of the Galois

_generators

of

_Dn-1,

for each valuation i in i

pn-leo -

1.

4.1.3. Previez,u:

_expressing

vZ . The observations in this subsection

depend

explicitly

upon

strong

ramificatzon,

and

_implicitly

_upon near maximal

rami-fication.

Note that near maximal ramification means that E

for _every cx E Hence E

_PÛn-1,

and so _z

must lie in the submodule of

_Dn-1/pDn-l

killed

_by

_{(~ -1).}

What are the

generators

of that submodule?

We turn to the _consequence of near maximal

_{ramification,}

_Proposition

3.5,

and the

_generators

of to address this

_question.

_Certainly

the elements

(~~’’ -1 )p-1 / (~ -1 ) ~

_~3

for

_~3

E

_Bj,t

where t

_~

0 lie in this module. Meanwhile the

_{elements $pj}

₍₎

for

_B3

E also lie in this module. So we

might

_expect

that vi

be

expressible

in terms of

However the

_4Dpi

_{(cr) ~}

_~3

for

_#

E _{are, in}

_turn,

_expressible

in terms of the Galois

_generators

of and so

_expressions

of the form are

not _necessary and _may be eliminated from this union.

To determine which of the members of

_(4.2)

_explicitly

_{appear in} the

description

of a

_{paxticular vi}

we make use of

_strong

_{ramification,.}

But first note that since elements in

_Dn-2

have

_valuation,

_equivalent

to

zero modulo _{p, it}

_might

seem that the residue class modulo _p of i should influence our

expression

for _vj. Indeed it

does;

our

expression

for Vi

will

depend initially

upon one of the

following

three conditions: i - 0 mod p, i -

_bl

mod _p, and

_{I fl 0,61}

mod _p.

_Ultimately,

it will

_depend

_upon a

partial

p-adic

expansion

for i.

Of the three

_conditions,

the last is most

_{easily explained. Using}

_strong

ramification

and its consequence Lemma

3.2,

Vi may be chosen to be

1)p_1/(p -

1)~3

for

_{some #}

E This _may be

accom-plished

without

_placing

_any additional restriction on

_~i

besides valuation. The condition i - 0

_{mod p}

is also

_{easily explained.}

Because of Lemma 2.1

there is an element of with

valuation,

equal

to i.

Therefore, by

induction we _may assume that there is an

_expression

in terms

of the Galois

_generators

for

_Dn-2

with valuation i. Use this

_expression

for

Vi

-This leaves the condition i -

_b1

mod p. Because of Lemma

3.2,

there is an element

_~3

E such that

_{1 )~-1 / (p -}

_)#

_{= p} + v where v has valuation i. So we _may

choose vi

:= v. We need to

_tidy

_up one loose end. What is the

_expression

in terms of Galois

_generators

for

_p?

We know

that p

E

Dn-2.

If we had an

expression

in

Dn-2

with the same valuation

as p,

(call

the

expression

p’),

then we could use Lemma

3.3,

to

replace ~3

with a new Galois

_generator,

~3’,

for and

replace va

with another

element,

vi,

of

_equivalent

valuation so

that vi

At that

_point,

we would have an element with valuation i which is

expressed

in terms of a new set of Galois

_generators

for

On-l-Of course, we still need to find an

_expression

_{among the} Galois

_generators

of with the same valuation as _p. As it turns out this

_expression

depends

on the residue class of its valuation. In

_fact,

there are three cases. As one

_{might imagine,}

when

bl

_{mod p,}

we will need to _express p itself in terms of a sum - one of these terms

_expressed

in terms of a Galois

_generator

for

_Dn-2/Dn-3’

the other term an element of

Dn-3.

At this

_point

it is clear that

_{the Vi}

could have a rather

_long

_expression

as a sum of

_generators

from different

_{quotient rings,}

the coefficients of which are of the form:

_{1 )p-1 / (p -}

₁₎

for

_{some j.}

4.2. The Recursive selection of the Galois

_generators.

Consider the

sequence with t-th term:

Recall that

_by

Remark 2.3 we _may assume that

bi

peo/(p 2013 1).

Since

peo - &#x3E;

_0,

it is

_easily

shown that the _sequence,

_{t = 1, ... },}

decreases as t increases with limit :=

pi-leo

+

_{bl -}

_{peo/(p -}

_1).

Therefore

_8oo(j)

_sl(j).

The

_significance

of this _sequence will be made clear later. For now note that under stable

_{ramification,}

0 for

j

In this

_section,

our

_goal

is the

_following:

To determine for each

_integer

i with i

_pi-leo

an

element vi

with valuation = i

_along

with its

_{explicit expression}

in terms of our

_generators

(i.e

an

explicit

linear combination of elements from

_along

with with coefficients from

t

For

_emphasis,

we state this

_again.

We would like to prove that:

Statement 4.2. For each

_integer

i with i

_pi-leo

there is an

element vi

E

_Ùj-1

with valuation

_Vj-1(Vi)

_{= i,}

and an

explicit

linear com-bination of elements from

_Ujk=oBk

with coefficients from

_{DT [G]}

that

_equals

~.

Clearly

Statement 4.2 holds

_{for j}

= 1: For each

integer

i with

i eo there is an element a of

Bo

with valuation

vo(a)

= i. To _prove

the Statement 4.2 holds

_{for j}

&#x3E; 1 and to

_provide

these

_explicit

linear

need to re-select certain elements of based _upon

_previous

selections

( $. e.

the elements of

4.2.1. Selection

_of

the elements

_of

_Bj,t.

For

_emphasis,

we

point

out that the elements in

_Bj,t

for t =

2,3,...

p -1

are chosen based _upon valuation alone. There is no need to

_replace

the elements

_given

in

_§3.

The first elements that we

replace

are those in

Bj,o .

We do so based _upon a subset of the elements

_{provided by}

Statement

_4.2,

_namely

those vi

E

with

_pi-1eo.

Note that We use Lemma 4.1.

Based upon vi we choose _paj,m E where m = i -

b1-

2pi-leo

+ eo. Is

is

_easily

checked that these bounds on i

_correspond

with the bounds on m

in Lemma 3.4.

Now,

we choose elements in based _upon

the vi

_{provided by}

State-ment 4.2 with i Our main tool is Lemma 3.3. Since the notation used in this lemma refers to the elements in as

_A’s,

we relabel our elements in

_referring

to them as _pj instead of _vi. So we

may say that we

_replace

elements of

_Bj,i

based _upon

the oi E

_given

by

Statement 4.2 with = i and i

sl(j).

Given a _pj, Lemma 3.3

_provides

elements a and v E

_Dj

such that

_{vj (a)}

=

bl

_{+ pm,}

=

pjeo

_{+ pm +}

bj,

and

where m = i -

bj .

In this _way the elements

_8j,1

are

defined for

_bj

For

_emphasis,

we

_repeat

this.

Only

those elements E

_8j,1

with

are redefined. All other elements in

_Bj-1

are left as

they

were in

§3.

One

may check that since

bl

&#x3E;

_{1/2~Peo/(p-1),}

so this

_really

refers to a subset of

We have defined all of the elements in

_Bj,o

and some of the elements in

_again

in terms of elements in In the

_following

lemma we show that the elements in which

_give

rise to

_Bj,o

are

disjoint

from the elements that

_give

rise to In other

_words,

_sl(j)

_Àj,j-1.

Lemma 4.3.

_bl

p

Proof.

From

_{(2.3) Àj,j-1}

= _{eo +}

Al,o.

This lemma is therefore

equivalent

to the statement that

_(p-1)bl

_pÀ1,0.

Since

_Al,o

=

bl -

Lb1/pJ,

4.2.2.

_Proof

that Statement 4.2 holds

_{with j}

instead

_{o f j -1.}

First consider those i - 0 mod _p with

_{pj eo}

+

_{b1 - peol(p -1)}

i

_{p~ eo.}

Let i =

p j,

then

(peo/(p - 1) - bl)

(peo/(p - 1) - bl)/p

j

and if Statement 4.2 holds

_{for j -1,}

then there are elements in

_Dj

with

_{valuation j expressed}

in terms of the Galois

_generators

of

The

_only

new

expressions

occur

when i =,4

0 mod _p. Based _upon Lemma

3.4,

as

long as i ~ 0, bl mod p,

there is an element

_{3

E so that

( ’ 1-1 ) p-1 IG -1

has valuation i. Observe that

1){3) :

,8

E is the set of all

_integers

_{0, bi mod p,}

_b1

i Because

_bl

&#x3E;

We have

_expressed

all

_{integers i #}

_bl

mod _p in the _range

_pjeo

+

_bl

-peo / ( p -1 )

i

_{p3 eo}

in terms of the Galois

_generators

for

_{D j .}

But we have not as

_yet

created _any new Galois

expressions

based _upon the old ones

provided by

Statement 4.2. Now consider i =

b1

_{mod p.}

One _may check that for each such

_integer

there are elements

_{3

E

_Bj,o

_{and p}

E such that the

_expression

_{for it}

in terms of the Galois

_generators

of

_Dj-1

is

_given,

and v =

(~~’ 1-

1)p-1/(p -1)~ -

p.

4.3.

_Description

of the Galois

_{relationships.}

Now that we have

re-cursively

defined the elements of fi to be

_compatible,

we describe their

relationships.

Given an

element vi

E

Dn-1

with

_{Vn-1(Vi) = i}

where

pn-leo,

we find that there are

_basically

two different _ways of

_{expressing vi}

in terms of the Galois

_generators

for

_On-

One of these

expressions

is in terms of the

_Bj,t

_{where t ~}

_0,

while the other

_expression

is in terms of the The

_complexity

of each

_expression

_depends

on the

p-adic expansion

of i.

4.3.1.

_{Expressing vi in}

terms

_of

the

_Bj,t

where t

_~

0. We

_proceed

now and describe the

_expression

for a

_{given vi}

E

Dn-i

with = i where First we examine the

p-adic

expansion

of i. We are interested in

only

certain of its

features,

and so we _express this

_p-adic

expansion

in a

_particular

_way.

Let b = Note b -

bl

mod

p.

Let b

be the least

nonnegative

residue of b modulo _p, and let A be the usual set of residues modulo p,

except b

is

_replaced

with b. So A :=

(10,

_{l, 2, ... ,}

_{p -}

1} -

f~1)

U{b}.

Each

integer

i &#x3E; 0 has a

_unique

finite

p-adic

representation

with coefficients in A:

Let m be the smallest

_subscript

with

c, 0

10, b}

or n -

_1,

whichever is smaller. In other

_words,

let