Submission date: 30 July 2014.

Abstract:

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

Keywords and phrases:

Full text arXiv 1407.8156: pdf, ps.