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Preprint Number 756
756. Franziska Jahnke and Jochen Koenigsmann
Uniformly defining p-henselian valuations
Submission date: 30 July 2014.
Admitting a p-henselian valuation is a weaker assumption on a field than admitting a 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.
Mathematics Subject Classification: Primary: 03C40, 12E30. Secondary: 12L12, 13J13, 16W60
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