Definable henselian valuations
<p style="text-align:justify;"> In this note we investigate the question when a henselian valued field carries a nontrivial ∅-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...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Published: |
Cambridge University Press
2015
|
| _version_ | 1826294394635943936 |
|---|---|
| author | Jahnke, F Koenigsmann, J |
| author_facet | Jahnke, F Koenigsmann, J |
| author_sort | Jahnke, F |
| collection | OXFORD |
| description | <p style="text-align:justify;"> In this note we investigate the question when a henselian valued field carries a nontrivial ∅-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 Prestel and Ziegler, there are further examples of henselian valued fields which do not admit a ∅-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 ∅-definable valuation when the residue field is separably closed or sufficiently nonhenselian, or when the absolute Galois group of the (residue) field is nonuniversal. </p> |
| first_indexed | 2024-03-07T03:44:57Z |
| format | Journal article |
| id | oxford-uuid:bf25fc8a-287c-4f08-91ea-6b4e20b7ae81 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T03:44:57Z |
| publishDate | 2015 |
| publisher | Cambridge University Press |
| record_format | dspace |
| spelling | oxford-uuid:bf25fc8a-287c-4f08-91ea-6b4e20b7ae812022-03-27T05:45:15ZDefinable henselian valuationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:bf25fc8a-287c-4f08-91ea-6b4e20b7ae81Symplectic Elements at OxfordCambridge University Press2015Jahnke, FKoenigsmann, J <p style="text-align:justify;"> In this note we investigate the question when a henselian valued field carries a nontrivial ∅-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 Prestel and Ziegler, there are further examples of henselian valued fields which do not admit a ∅-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 ∅-definable valuation when the residue field is separably closed or sufficiently nonhenselian, or when the absolute Galois group of the (residue) field is nonuniversal. </p> |
| 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 |