Defining coarsenings of valuations

TL;DR

This paper investigates which henselian fields have definable henselian valuations, providing characterizations in certain cases and exploring the Galois-theoretic implications of these definability properties.

Contribution

It offers a complete characterization of henselian fields with parameter-definable valuations in equicharacteristic zero and partial results in other cases.

Findings

Fields with non-divisible value groups admit parameter-definable henselian valuations.

Complete characterization in equicharacteristic zero.

Partial characterization of 0-definable henselian valuations.

Abstract

We study the question which henselian fields admit definable henselian valuations (with or without parameters). We show that every field which admits a henselian valuation with non-divisible value group admits a parameter-definable (non-trivial) henselian valuation. In equicharacteristic $0$, we give a complete characterization of henselian fields admitting a parameter-definable (non-trivial) henselian valuation. We also obtain partial characterization results of fields admitting 0-definable (non-trivial) henselian valuations. We then draw some Galois-theoretic conclusions from our results.