Definable henselian valuations
In this note we investigate the question whether a henselian valued field carries a non-trivial 0-definable henselian valuation (in the language of rings). It follows from the work of Prestel and Ziegler that there are henselian valued fields which do not admit a 0-definable non-trivial henselian va...
| Main Authors: | , |
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| Format: | Journal article |
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2012
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| Summary: | In this note we investigate the question whether a henselian valued field carries a non-trivial 0-definable henselian valuation (in the language of rings). It follows from the work of Prestel and Ziegler that there are henselian valued fields which do not admit a 0-definable non-trivial henselian valuation. We give conditions on the residue field which ensure the existence of a parameter-free definiton. In particular, we show that a henselian valued field admits a non-trivial 0-definable valuation when the residue field is separably closed or sufficiently non-henselian, or when the absolute Galois group of the (residue) field is non-universal. |
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