An existential ∅-definition of Fq[[t]]Fq[[t]] IN Fq(t)
<p style="text-align:justify;">We show that the valuation ring Fq [[t]] in the local field Fq ((t)) is existentially definable in the language of rings with no parameters. The method is to use the definition of the henselian topology following the work of Prestel-Ziegler to give an...
| Autori principali: | , |
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| Natura: | Journal article |
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Cambridge University Press
2014
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| _version_ | 1826306551929896960 |
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| author | Anscombe, W Koenigsmann, J |
| author_facet | Anscombe, W Koenigsmann, J |
| author_sort | Anscombe, W |
| collection | OXFORD |
| description | <p style="text-align:justify;">We show that the valuation ring Fq [[t]] in the local field Fq ((t)) is existentially definable in the language of rings with no parameters. The method is to use the definition of the henselian topology following the work of Prestel-Ziegler to give an ∃-Fq -definable bounded neighbouhood of 0. Then we “tweak” this set by subtracting, taking roots, and applying Hensel’s Lemma in order to find an ∃-Fq -definable subset of Fq [[t]] which contains tFq [[t]]. Finally, we use the fact that Fq is defined by the formula xq − x = 0 to extend the definition to the whole of Fq [[t]] and to rid the definition of parameters. Several extensions of the theorem are obtained, notably an ∃-∅-definition of the valuation ring of a nontrivial valuation with divisible value group.</p> |
| first_indexed | 2024-03-07T06:49:40Z |
| format | Journal article |
| id | oxford-uuid:fc1a93b2-cf61-463d-80a2-9bec32a29d3d |
| institution | University of Oxford |
| last_indexed | 2024-03-07T06:49:40Z |
| publishDate | 2014 |
| publisher | Cambridge University Press |
| record_format | dspace |
| spelling | oxford-uuid:fc1a93b2-cf61-463d-80a2-9bec32a29d3d2022-03-27T13:18:23ZAn existential ∅-definition of Fq[[t]]Fq[[t]] IN Fq(t)Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:fc1a93b2-cf61-463d-80a2-9bec32a29d3dSymplectic Elements at OxfordCambridge University Press2014Anscombe, WKoenigsmann, J <p style="text-align:justify;">We show that the valuation ring Fq [[t]] in the local field Fq ((t)) is existentially definable in the language of rings with no parameters. The method is to use the definition of the henselian topology following the work of Prestel-Ziegler to give an ∃-Fq -definable bounded neighbouhood of 0. Then we “tweak” this set by subtracting, taking roots, and applying Hensel’s Lemma in order to find an ∃-Fq -definable subset of Fq [[t]] which contains tFq [[t]]. Finally, we use the fact that Fq is defined by the formula xq − x = 0 to extend the definition to the whole of Fq [[t]] and to rid the definition of parameters. Several extensions of the theorem are obtained, notably an ∃-∅-definition of the valuation ring of a nontrivial valuation with divisible value group.</p> |
| spellingShingle | Anscombe, W Koenigsmann, J An existential ∅-definition of Fq[[t]]Fq[[t]] IN Fq(t) |
| title | An existential ∅-definition of Fq[[t]]Fq[[t]] IN Fq(t) |
| title_full | An existential ∅-definition of Fq[[t]]Fq[[t]] IN Fq(t) |
| title_fullStr | An existential ∅-definition of Fq[[t]]Fq[[t]] IN Fq(t) |
| title_full_unstemmed | An existential ∅-definition of Fq[[t]]Fq[[t]] IN Fq(t) |
| title_short | An existential ∅-definition of Fq[[t]]Fq[[t]] IN Fq(t) |
| title_sort | existential ∅ definition of fq t fq t in fq t |
| work_keys_str_mv | AT anscombew anexistentialdefinitionoffqtfqtinfqt AT koenigsmannj anexistentialdefinitionoffqtfqtinfqt AT anscombew existentialdefinitionoffqtfqtinfqt AT koenigsmannj existentialdefinitionoffqtfqtinfqt |