Uniformly defining p-henselian valuations

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 closed nor real closed.