A Choice-free look at algebraic extensions of valuations

TL;DR

This paper investigates algebraic valuation extensions without relying on the Axiom of Choice, establishing a bijection with maximal ideals and providing an elementary proof of the fundamental inequality.

Contribution

It introduces a choice-free framework for understanding valuation extensions and proves the existence of maximal ideals in finite extensions.

Findings

Bijection between valuation extensions and maximal ideals

Existence of maximal ideals in finite extensions

Elementary proof of the fundamental inequality

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.