Key polynomials, separate and immediate valuations, and simple extensions of valued fields

TL;DR

This paper introduces a first-order definition of key polynomials and explores their connections with valuations, key degrees, and partially multiplicative valuations, providing new insights into simple extensions of valued fields.

Contribution

It offers a novel first-order framework for key polynomials and clarifies their relationship with valuations and key degrees in valued field extensions.

Findings

Defined key polynomials in a first-order manner

Linked key polynomials with partially multiplicative valuations

Reproved several properties of key polynomials and valuations

Abstract

We give a first-order definition of key polynomials, we show the links with previous definitions, that it is relevant to study key degrees, and to use a kind of valuations that we call partially multiplicative. We also prove or reprove several properties.