On Residually Generic Prolongations of a Valuation to a Simple Transcendental Extension

Abstract

Throughout the paper K=K₀(x) denotes a simple transcendental extension of a field K₀ν₀ a Krull valuation of K₀, and ν a Prolongation of ν₀ to K. Also A₀⊆A. K₀⊆k and G₀⊆G denote respectively the valuation rings, residue fields and value groups of the Valuations ν₀ and ν. For an element ξ in A, ξ* will stand for its image in the residue field k of ν. We shall some times refer to ξ* as the ν-residue of ξ.