On common extensions of valued fields

TL;DR

This paper investigates the structure of common valuation extensions from a base field to its algebraic closure and rational function fields, establishing links between key polynomials and minimal pairs, and characterizing maximal sequences and roots in transcendental cases.

Contribution

It introduces a detailed connection between minimal pairs and key polynomials, and characterizes the existence of common extensions in transcendental cases.

Findings

A link between minimal pairs and key polynomials is established.

Any sequence of key polynomials has a maximal element in transcendental extensions.

Roots of the last key polynomial describe common extensions when no limit key polynomial exists.

Abstract

Given a valuation $v$ on a field $K$, an extension $\bar{v}$ to an algebraic closure and an extension $w$ to $K(X)$. We want to study the common extensions of $\bar{v}$ and $w$ to $\bar{K}(X)$. First we give a detailed link between the minimal pairs notion and the key polynomials notion. Then we prove that in the case when $w$ is a transcendental extension, then any sequence of key polynomials admits a maximal element, and in case this sequence does not contain a limit key polynomial, then any root of the last key polynomial, describe a common extension.