8 GUILLAUME DUVAL

Definition 9. Let F/K be a field extension and G a subgroup of Aut(F/K).

G is said to have a locally-finite action on S(F/K)∗ if for all ν in S(F/K)∗, for all

x in F , {ν(σ(x))|σ ∈ G} is a finite subset of Γν.

Convention: Throughout this paper, if Ω is a ring, an ideal of a ring, or, a

field, we set

Ω∗

= Ω\ {0}.

Proposition 10. Let F/K be a field extension and G a subgroup of Aut(F/K)

with a locally-finite action on

S(F/K)∗.

Then for any valuation ν of F/K, the

following three assertions are equivalent:

i. The valuation ν is strongly G-invariant.

ii. For all x in F ∗ and all σ in G,

σ(x)

x

belongs to U(Rν ).

iii. The valuation ν is G-invariant.

Although (i) and (ii) are equivalent, they are distinct in nature. Condition

(i) expresses the strong invariance in terms of the valuation, and (ii) in terms of

the valuation ring. For this reason we shall speak later of a strongly G-invariant

valuation ring.

Proof. Since strong invariance implies invariance, it still needs to be shown

that (iii) implies (i). Assume that (iii) is true. Then, every σ ∈ G induces an

automorphism of the local ring Rν, therefore σ(mν) = mν and σ(U(Rν)) = U(Rν).

Hence, for all x ∈ F

∗

and all σ ∈ G, ν(σ(x)) and ν(x) are either both strictly

positive, or zero, or both negative in Γν. Let us assume, aiming for contradiction,

that (i) does not hold. This means that we would have ν(σ(x)) = ν(x) for some x

in F

∗

and some σ ∈ G. Replacing x by

x−1

if necessary, we can assume without

loss of generality that ν(σ(x)) ν(x).

Since ν(σ(x)) ν(x), there exists Q1 ∈ mν such that σ(x) = Q1 · x. Now, set

σn(x)

= Qn · x for all n 1. We therefore obtain the following recursive formulae:

σn+1(x)

=

σn

◦ σ(x) =

σn(Q1)

·

σn(x)

=

σn(Q1)

· Qn · x

= Qn+1 · x.

Since Q1 belongs to mν,

σn(Q1)

also belongs to mν. Therefore

ν(Qn+1) =

ν(σn(Q1))

+ ν(Qn) ν(Qn).

Therefore the sequence

{ν(σn(x))}n∈

is strictly increasing in Γν , hence, cannot

be finite. This contradicts the local finiteness assumption.

Remark 11. The mainspring behind the latter argument is the following: Let’s

assume that ν is G-invariant. Every σ ∈ G induces an order preserving and additive

automorphism ¯ σ on the value group Γν. Here, ¯ σ is uniquely determined by the

formula

¯(ν(x)) σ = ν(σ(x)).

Denoting by

Aut+(Γν)

the whole group of order preserving-additive automorphisms

of Γν, we are led to a group morphism

φν : G −→

Aut+(Γν).

The invariant valuation ν will be strongly G-invariant if and only if φν is trivial. The

proof of Proposition 10 asserts that every non-trivial element ϕ ∈

Aut+(Γν)

must