Henselianity in the language of rings

TL;DR

This paper investigates four properties related to definable henselian valuations in fields, revealing that their relationships differ significantly between equicharacteristic and mixed characteristic cases.

Contribution

It provides a detailed analysis of the implications between properties of henselian valuations and highlights the differences across characteristic types.

Findings

Implication structures vary between equicharacteristic and mixed characteristic fields.

Certain properties are equivalent in one characteristic setting but not in the other.

The study clarifies the landscape of definable henselian valuations in different field contexts.

Abstract

We consider four properties of a field $K$ related to the existence of (definable) henselian valuations on $K$ and on elementarily equivalent fields, and study the implications between them. Surprisingly, the full pictures look very different in equicharacteristic and mixed characteristic.