Separable extension

Abstract: We want to determine all the extensions of a valuation v of a field K to a cyclic extension L of K, i.e. L = K(x) is the field of rational functions of x or L = K(θ) is the finite separable extension generated by a root 0 of an irreducible polynomial 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 of K[x], and has shown how we can recover any extension to L of a discrete rank one valuation v of K by a countable sequence of augmented valuations (μ ι ) ι ∈ with I ⊂ N. The valuation μ i is defined by induction from the valuation μ i-1 , from a key polynomial o i and from the value γ, = μ(oi). In this article we study some properties of the augmented valuations and we generalize the results of MacLane to the case of any valuation v of K. For this we need to introduce simple admissible families of augmented valuations A (μα) α ∈A where 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 to L of a valuation ν on 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)) for any polynomial f in K [x] .