A ruled residue theorem for function fields of conics

TL;DR

This paper extends the ruled residue theorem to function fields of conics, characterizing valuation extensions and their residue field extensions, especially focusing on cases with transcendental but not ruled residue extensions.

Contribution

It generalizes the ruled residue theorem to conic function fields and characterizes the unique valuation extension with transcendental non-ruled residue extension.

Findings

At most one valuation extension has a transcendental non-ruled residue field extension.

The paper characterizes when this special valuation extension occurs.

Provides a comprehensive understanding of valuation extensions in conic function fields.

Abstract

The ruled residue theorem characterises residue field extensions for valuations on a rational function field. Under the assumption that the characteristic of the residue field is different from $2$ this theorem is extended here to function fields of conics. The main result is that there is at most one extension of a valuation from on the base field to the function field of a conic for which the residue field extension is transcendental but not ruled. Furthermore the situation when this valuation is present is characterised.