Definable non-divisible Henselian valuations. (English) Zbl 1319.03049

Summary: On a Henselian valued field \((K, V)\), where \(V\) is the valuation ring, if the value group contains a convex \(p\)-regular subgroup that is not \(p\)-divisible, then \(V\) is definable in the language of rings. A Henselian valuation ring with a regular non-divisible value group is always 0-definable. In particular, some results of J. Ax [Proc. Am. Math. Soc. 16, 846 (1965; Zbl 0199.03003)] and of J. Koenigsmann [Sib. Adv. Math. 14, No. 3, 16–42 (2004; Zbl 1074.12003)] are generalized.


03C40 Interpolation, preservation, definability
12E30 Field arithmetic
12J10 Valued fields
03C60 Model-theoretic algebra
12L12 Model theory of fields
13F30 Valuation rings
13J15 Henselian rings
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