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Definable Henselian valuations. (English) Zbl 1372.03078

Summary: In this note we investigate the question when a Henselian valued field carries a nontrivial \(\emptyset\)-definable Henselian valuation (in the language of rings). This is clearly not possible when the field is either separably or real closed, and, by the work of A. Prestel and M. Ziegler [J. Reine Angew. Math. 299/300, 318–341 (1978; Zbl 0367.12014)], there are further examples of Henselian valued fields which do not admit a \(\emptyset\)-definable nontrivial Henselian valuation. We give conditions on the residue field which ensure the existence of a parameter-free definition. In particular, we show that a Henselian valued field admits a nontrivial Henselian \(\emptyset\)-definable valuation when the residue field is separably closed or sufficiently nonhenselian, or when the absolute Galois group of the (residue) field is nonuniversal.

MSC:

03C60 Model-theoretic algebra
03C40 Interpolation, preservation, definability
12J10 Valued fields
12L12 Model theory of fields

Citations:

Zbl 0367.12014

References:

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