Uniformly defining $p$-henselian valuations

TL;DR

This paper classifies when the canonical $p$-henselian valuation on a field is uniformly 0-definable, and demonstrates the existence of a definable valuation inducing the henselian topology on certain fields.

Contribution

It provides a classification of elementary classes of fields with uniformly 0-definable $p$-henselian valuations and constructs definable valuations inducing henselian topologies.

Findings

Classification of elementary classes with uniform 0-definability

Existence of definable valuations inducing henselian topology

Application to non-separably and non-real closed fields

Abstract

Admitting a non-trivial $p$-henselian valuation is a weaker assumption on a field than admitting a non-trivial henselian valuation. Unlike henselianity, $p$-henselianity is an elementary property in the language of rings. We are interested in the question when a field admits a non-trivial 0-definable $p$-henselian valuation (in the language of rings). We give a classification of elementary classes of fields in which the canonical $p$-henselian valuation is uniformly 0-definable. We then apply this to show that there is a definable valuation inducing the ($t$-)henselian topology on any ($t$-)henselian field which is neither separably nor real closed.