Tame Key polynomials

TL;DR

This paper presents a new method for constructing complete sequences of key polynomials for tame field extensions, linking algebraic and transcendental cases through minimal polynomials and implicit constant fields.

Contribution

Introduces a novel approach to constructing key polynomials using minimal polynomials over the base field, applicable to both algebraic and transcendental tame extensions.

Findings

Method works for simple algebraic tame extensions

Extension to countably generated infinite tame extensions

Highlights the role of implicit constant fields in transcendental cases

Abstract

We introduce a new method of constructing complete sequences of key polynomials for simple extensions of tame fields. In our approach the key polynomials are taken to be the minimal polynomials over the base field of suitably constructed elements in its algebraic closure, with the extensions generated by them forming an increasing chain. In the case of algebraic extensions, we generalize the results to countably generated infinite tame extensions over henselian but not necessarily tame fields. In the case of transcendental extensions, we demonstrate the central role that is played by the implicit constant fields, which reveals the tight connection with the algebraic case.