comes from a separable field embedding of
proposition follows from Prop.7. Therefore, by Prop.ll we can assume that
isomorphism and we must prove that for any
it holds p*(r~l)
Suppose that L' is a finite Galois extension of L, then K'
P(L') is Galois
over K and we obtain induced group isomorphism
L' I L
K' I K.
verify that for any
J ( n),
The compatibility of
implies that for all 0
is compatible with natural projections r
L' I L
K' I K
K' . IK . ,
and induces group isomorphisms
We can assume by induction that
are compatible with ramification filtration
n and, therefore, it is sufficient to prove (8) only for all
E Jn. Notice
also that ( 9) implies that e( L'
Choose a finite extension M of L~,n- 1 such that if
L' M, L
a) there is L~ with standard F-structure such that L'
b) there is K~ with standard F -structure such that K'
Then there are
E L~ and
E K~ such that
Remark in n.4. Therefore,
c,n-1 c,n 1 c,n 1
If VL' and vK' are valuations of rank n from Prop.3, then p*vL' =
VK', i.e. for any z
PROOF OF LEMMA. Because
K'), it will be sufficient to prove
By induction we can assume also that
when being restricted to K~,n- 1 .
Notice, that any system of local parameters of K~,n-
being completed by
gives a system of local parameters of
So, we must prove only that
From the definition of valuations
ey) 1 L~ V£~ (OL)
(0, ... , 0, 1) and, similarly,
(0, ... , 0, 1), i.e.
Vf! (OK) =VI,, (OL)·
remains only to note that
appear as lifts of uniformizing elements
of the (n- 1)-th residue fields of the fields
are isomorphic under
The lemma is proved.
From (10) it follows that we can use
to compute rami-
fication invariants of
So, for any
V£1 ( r( p(OK))- p( OK)) -VL' ( p(OK))
VK' ( p* ( r)OK -OK) -VK' (OK)
iK' 1 K( p*T ).
The proposition is proved.