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

Let K 0 (x) be a simple transcendental extension of a field K 0 , υ 0 be a valuation of K 0 with value group G 0 and residue field K 0 . SupposeG 0 ⊆ G 1 ⊆ G is an inclusion of totally ordered abelian groups with [ G 1 : G 0 ] < ∞ such that G is the direct sum of G 1 and an infinite cyclic group. It is proved that there exists an (explicitly constructible) valuation υ of K 0 ( x ) extending υ 0 such that the value group of υ is G and its residue field is k , where k is a given finite extension of k 0 . This is analogous to a result of Matignon and Ohm [2, Corollary 3.2] for residually non‐algebraic prolongations of υ 0 to K 0 ( x ).