8 VICTOR ABRASHKIN

Indeed,

ZEIL("Yo8)

=

VE(("Yo8)B- B)- VE(B)

~

min{vEbo(8B- B)),vEboB- B)}- VE(B)

=

min{i'EIL(8),i'EILbo)},

and this inequality becomes the equality ifi' E 1 L( 8)

i'

E 1 L( 'Yo). On the other hand,

if i'EI£(8)

~

i'Eid"Yo), then

d(w)

~

i'Eid"Yo8)

~

min{i'EI£(8), i'Eid"Yo)}

=

i'Eid"Yo)

=

d(w)

and the equality ( 4) still holds. So,

'PEIF(d(w))

=

e-p;jF

L

min{i'EIF(8), d(w)}

=

e£:/F

L

i'Eid"Y)

(notice that i'EIF(8)

=

i'EI£(8)) and our proposition will be implied by the following

lemma.

LEMMA.

For any

wE

fElL, it

holds

· eEIFi'FIL(w)

=

L

i'Eid"Y)·

')'EfE/L

7r("Y)=w

PROOF.

As earlier, we have

Off;

=

O£[B] and

OF

=

O£[B'], where

B'

NEIF(B). Let

f(X)

=

xm

+

a1Xm-l +···+am

E

Offo[X]

be the minimal monic polynomial of B over F. Consider

(wf)(X)

=

xm

+

w(al)xm-l

+ · · · +

w(am)

E

Offo[X].

Clearly, am= ( -l)mB', i'p1L(w)

+

vp(B')

=

vp(wam- am) vp((wan- an)Bm-n)

for all 1

~

n m, and therefore,

VE((wf)(B)- j(B))

=

eEIFVF((wf)(B)- f(B))

=

eEIF(i'FIL(w)

+

vp(B')).

On the other hand, the equality

implies

(wf)(B)- f(B)

=

IT

(B -')'B)

')'EfE/L

7r("Y)=w

vE((wf)(B)- f(B))

=

L

(iEid"Y)

+

VE(B))

')'EfE/L

7r("Y)=w

and it remains to notice that eEIFVF(B')

=

VE(B')

=

[E:

F]vE(B).

PROPOSITION

5.

For any j

E

Jn, it holds

(5)