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) = 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 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
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