Prolongations of valuations to simple transcendental extensions with given residue field and value group

Abstract

Let K₀(x) be a simple transcendental extension of a field K₀, υ₀ be a valuation of K₀ with value group G₀ and residue field K₀. Suppose G₀⊆G₁⊆G is an inclusion of totally ordered abelian groups with [G₁: G₀] <∞such that G is the direct sum of G₁ and an infinite cyclic group. It is proved that there exists an (explicitly constructible) valuation υ of K₀(x) extending υ₀ such that the value group of υ is G and its residue field is k, where k is a given finite extension of k₀. This is analogous to a result of Matignon and Ohm [2, Corollary 3.2] for residually non-algebraic prolongations of υ₀ to K₀(x).