16 GUILLAUME DUVAL
possible to extend the action from G(C) to G(K), this latter must be given by the
corresponding relations
(RK) :˜(z) σ =
d
i=1
ai(˜)zi, σ ∀˜ σ G(K).
In the particular case where K is algebraically closed, the Kolchin-Singer’s Theorem
(Proposition 15(v)) states precisely that (RK) allows us to extend the action to
G(K) and this group can be viewed as an automorphism group of F/K.
The general case can be deduced from the previous one, thanks to Proposi-
tion 15(vi). This concludes the description of the action of
G◦(K)
on F by field
automorphisms.
(iii) Since the considered valuations ν are trivial on K, we may replace C by
K in Lemma 18, and from relations (RK) we still get
∀z T (F/K), card{ν(˜(z))|˜ σ σ
G◦(K)}
ordK (z).
Therefore, as in the proof of (i),
G◦(K)
has a locally-finite action on
S(F/K)∗.
Now, let ν be a non-trivial G(C)-invariant valuation of F/K. By Proposition
10, ν is strongly G(C)-invariant and for all z T (F/K), we have ν(z1) = ν(z2) =
· · · =
ν(zd), where {z = z1,...,zd} is a C-basis of V (z) formed by conjugates of z
under G(C). Therefore, from formulae (RK), we have
∀˜ σ
G◦(K),ν
(˜(z)) σ = ν
d
i=1
ai(˜)zi σ
inf{ν(zi),i = 1,...,d}
∀˜ σ
G◦(K),ν
(˜(z)) σ ν(z).
Since ˜(z) σ still belongs to T (F/K) by Proposition 15(i) and since
G◦(K)
is a group,
from the previous inequality we deduce the more precise one
∀˜ σ
G◦(K),ν(z)
= ν
˜−1(σ(z))
σ ν (˜(z)) σ ν (z) .
Therefore, equality holds in the latter formula, and ν is strongly G◦(K)-invariant.
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