On valuations of K(X)

Abstract

For a valued field (K, v), let K_v denote the residue field of v and G_v its value group. One way of extending a valuation v defined on a field K to a simple transcendental extension K(x) is to choose any α in K and any µin a totally ordered Abelian group containing G_v, and define a valuation w on K[x] by w(Σ_ic_i(x−α)ⁱ)=min_i.(ν (c_i)+iµ.Clearly either G_v is a subgroup of finite index in G_W=G_v + Zµ or G_W/G_v is not a torsion group. It can be easily shown that K(x)_w is a simple transcendental extension of K_v in the former case. Conversely it is well known that for an algebraically closed field K with a valuation ν, if w is an extension of ν to K(x) such that either K(x)_w is not algebraic over K_v or G_w/G_v is not a torsion group, then w is of the type described above. The present paper deals with the converse problem for any field K. It determines explicitly all such valuations w together with their residue fields and value groups.