2014/07/30 by Jahnke, Franziska, Koenigsmann, Jochen
#12E30. Secondary: 12L12 #13J13 #16W60 #Commutative Algebra (math.AC) #FOS: Mathematics #Logic (math.LO) #Primary: 03C40
paper · doi:10.48550/arxiv.1407.8156
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.