Uniformly defining $p$-henselian valuations

Franziska Jahnke; Jochen Koenigsmann

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arXiv · 1407.8156

Uniformly defining $p$-henselian valuations

Franziska Jahnke·Jochen Koenigsmann

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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.

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BibTeX RIS

Franziska Jahnke, Jochen Koenigsmann. 2014-07-30. Uniformly defining $p$-henselian valuations. https://arxiv.org/abs/1407.8156

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