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)
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