Value groups, residue fields and bad places of rational function fields

TL;DR

This paper classifies all valuation extensions from a base field to rational function fields, detailing possible value groups and residue fields, and provides constructions for extensions with specific properties, including non-finitely generated cases.

Contribution

It offers a comprehensive classification of valuation extensions in rational function fields and introduces methods to construct extensions with prescribed value groups and residue fields.

Findings

Extensions can have non-finitely generated value groups and residue fields.

The relative algebraic closure in henselizations can be any countably generated separable extension.

Methods are provided for power series fields and p-adic fields.

Abstract

We classify all possible extensions of a valuation from a ground field $K$ to a rational function field in one or several variables over $K$. We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. In the case of a rational function field $K(x)$ in one variable, we consider the relative algebraic closure of $K$ in the henselization of $K(x)$ with respect to the given extension, and we show that this can be any countably generated separable-algebraic extension of $K$. In the "tame case", we show how to determine this relative algebraic closure. Finally, we apply our methods to power series fields and the $p$-adics.