Value groups and simple transcendental extensions

Abstract

Throughout the paper K(x) is a simple transcendental extension of a field K; v is a valuation of K and w is an extension of v to K(x).Also k_O Ík and G_OÍG denote respectively the residue fields and the value groups of the valuations v and w. A well-known theorem conjectured by Nagata asserts that either k is an algebraic extension of feo or k is a simple transcendental extension of a finite extension of k_O(cf [4] or [6] or [1, Corollary 2.3]). We prove here an analogous result for the value groups viz. either G/ Gk_O s a torsion group or there exists a subgroup G1 of G containing Go with [G1: Go] > ∞ such that G is the direct sum of G1 and an infinite cyclic group. Incidentally we obtain a description of the valuation w as well as of its residue field in the second case.Thus a characterization of all those extensions w of v to K(x), for which w(K(x)\{0})/Go is not a torsion group, is given. Corresponding to such a valuation w, we define three numbers N, S and T which satisfy the inequality N ≥= ST. This is analogous to the fundamental inequality established by Ohm (cf. [5, 1.2]) for residually transcendental extensions of v to K(x). We also investigate the conditions under which N = ST.