A Choice-free look at algebraic extensions of valuations

Abstract

In this paper, we study extensions of valuations over algebraic field extensions without the use of the Axiom of Choice. We show a bijection between the extensions of a valuation and the maximal ideals of the relative integral closure of its valuation ring. In the case of a finite extension, we show that these maximal ideals exist. We conclude with an elementary proof of the fundamental inequality.