Definable henselian valuations

TL;DR

This paper explores conditions under which henselian valued fields have a non-trivial 0-definable henselian valuation, providing criteria based on residue field properties and Galois group characteristics.

Contribution

It establishes new conditions on residue fields that guarantee the existence of a parameter-free definable henselian valuation.

Findings

Henselian fields with separably closed residue fields admit 0-definable valuations.

Fields with certain non-henselian residue fields also admit such definitions.

Non-universal Galois groups of residue fields imply definability of valuations.

Abstract

In this note we investigate the question whether a henselian valued field carries a non-trivial 0-definable henselian valuation (in the language of rings). It follows from the work of Prestel and Ziegler that there are henselian valued fields which do not admit a 0-definable non-trivial henselian valuation. We give conditions on the residue field which ensure the existence of a parameter-free definiton. In particular, we show that a henselian valued field admits a non-trivial 0-definable valuation when the residue field is separably closed or sufficiently non-henselian, or when the absolute Galois group of the (residue) field is non-universal.