Extension d’une valuation

Abstract

We want to determine all the extensions of a valuation ν \nu of a field K K to a cyclic extension L L of K K , i.e. L = K ( x ) L=K(x) is the field of rational functions of x x or L = K ( θ ) L=K(\theta ) is the finite separable extension generated by a root θ \theta of an irreducible polynomial G ( x ) G(x) . In two articles from 1936, Saunders MacLane has introduced the notions of key polynomial and of augmented valuation for a given valuation μ \mu of K [ x ] K[x] , and has shown how we can recover any extension to L L of a discrete rank one valuation ν \nu of K K by a countable sequence of augmented valuations ( μ i ) i ∈ I \bigl (\mu _i\bigr ) _{i \in I} , with I ⊂ N I \subset \mathbb N . The valuation μ i \mu _i is defined by induction from the valuation μ i − 1 \mu _{i-1} , from a key polynomial ϕ i \phi _i and from the value γ i = μ ( ϕ i ) \gamma _i = \mu ( \phi _i ) . In this article we study some properties of the augmented valuations and we generalize the results of MacLane to the case of any valuation ν \nu of K K . For this we need to introduce simple admissible families of augmented valuations A = ( μ α ) α ∈ A {\mathcal A} = \bigl ( \mu _{\alpha } \bigr ) _{\alpha \in A} , where A A is not necessarily a countable set, and to define a limit key polynomial and limit augmented valuation for such families. Then, any extension μ \mu to L L of a valuation ν \nu on K K is again a limit of a family of augmented valuations. We also get a “factorization” theorem which gives a description of the values ( μ α ( f ) ) ( \mu _{\alpha } (f)) for any polynomial f f in K [ x ] K[x] .