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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| _version_ | 1797056205728776192 |
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| author | Jahnke, F Koenigsmann, J |
| author_facet | Jahnke, F Koenigsmann, J |
| author_sort | Jahnke, F |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-06T19:20:04Z |
| format | Journal article |
| id | oxford-uuid:19c4390f-009e-4e05-b7b7-fac9ee99f22c |
| institution | University of Oxford |
| last_indexed | 2024-03-06T19:20:04Z |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oxford-uuid:19c4390f-009e-4e05-b7b7-fac9ee99f22c2022-03-26T10:50:51ZDefinable henselian valuationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:19c4390f-009e-4e05-b7b7-fac9ee99f22cSymplectic Elements at Oxford2012Jahnke, FKoenigsmann, JIn 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. |
| spellingShingle | Jahnke, F Koenigsmann, J Definable henselian valuations |
| title | Definable henselian valuations |
| title_full | Definable henselian valuations |
| title_fullStr | Definable henselian valuations |
| title_full_unstemmed | Definable henselian valuations |
| title_short | Definable henselian valuations |
| title_sort | definable henselian valuations |
| work_keys_str_mv | AT jahnkef definablehenselianvaluations AT koenigsmannj definablehenselianvaluations |