4

VICTOR ABRASHKIN

of "0-dimensional constants" Le o. In the mixed characteristic case L contains

Qp,

therefore, Le 1 is the algebraic closure of

Qp

in L, and we take its maximal

unramified over

Qp

subfield as Leo.

If

E is a finite extension of L then E is provided with a unique induced F-

structure such that for any 1

~

i

~

n, Ee i is the algebraic closure of Lei in

E. Inversely, any given F-structure onE induces the F-structure of L given by its

subfields Lei

:=

LnEei· In the both cases above we shall call F-structures of E and

L

compatible. Throughout all this paper all local fields are assumed to be provided

with some

F

-structure. When considering any algebraic extension of n-dimensional

local fields we always assume that the corresponding F -structures are compatible.

Notice also that for 1

~

i

~

n, the subfields

L~~]_

1

:=

(Lei

n 0}_1))

mod

m}_1l

give

the induced F-structure of the first residue field

L(1)

of

L.

So, while giving an

F-structure on L we provide automatically all residue fields of L with uniquely

determined induced F-structures.

Suppose

L

is a standard field. Then either

L

=

k((tn)) ...

((t1)), where

k

is

a finite field, or

L = K { {

tn}} ... { { t2} }, where

K

is a !-dimensional local field

with uniformizing element t1. In the both cases t1, ... , tn form a system of local

parameters in L. Associate to it the F-structure of L such that for

1

~

j

~

n, the

subfield Lej consists of elements l given in terms of corresponding formal series

(2)

by the condition

O:i

1

...

in

=

0 if at least one of the indices iH1, ... , in is not zero.

In other words, for 1

~

j

~

n, the subfield Lej consists of elements presented as

formal series in variables t1, ... , tj. This F-structure of (a standard field) L will be

called standard. The following proposition is very well-known application of Epp's

result on eliminating wild ramification.

PROPOSITION

1. Let L be an n-dimensional local field with F -structure. Then

there is a finite separable extension E' of Le,n-1 such that the induced F-structure

on E

:=

LE' is standard.

PROOF.

Apply induction on n.

If n = 1

then there is nothing to prove.

Let n

1

and let

Lalg

be an algebraic closure of

L.

By Epp's Theorem [Epp,

KZ] there is a finite separable extension M1 of Le 1 in

Lalg

such that if M

=

LM1,

then any uniformizing element t1 of M1 appears also as a uniformizing element of

M

(with respect to its first valuation). Let M'

=

M(1)

be the first residue field of M.

Consider its induced F- structure {M~i 11 ~

i

~ n- 1}, where M~i

=

(Me,i+1)(1)

is the first residue field of Me,i+1· By induction there is a finite separable extension

E'

of

M~,n- 2

in

L~~

such that the induced F- structure

{Eei

I

1

~

i

~

n- 1}

of

E

:=

E'

M' is standard, i.e. it is associated to some system of local parameters

f1, ... , tn-1 of

E.

Let E be one of unramified extensions of Min

La1g

with the (first) residue field

E.

For 1

~ i ~

n, denote by Eei the maximal unramified extension of Mei in E.

It is easy to see that {Eei

I

1

~

i

~

n} is F-structure on E, this F-structure is

associated to a collection of local parameters t1, ... ,

tn

such that for

i = 2, ... ,

n,

ti

E

OE(

1)

and ti mod

mE(

1)

=

fi-1· The proposition is proved.

c• c'