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 0367.12014References:
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