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.