Model completeness for Henselian fields with finite ramification valued in a $Z$-group
We prove that the theory of a Henselian valued field of characteristic zero, with finite ramification, and whose value group is a $Z$-group, is model-complete in the language of rings if the theory of its residue field is model-complete in the language of rings. We apply this to prove that every inf...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Published: |
Cornell University
2016
|
| _version_ | 1797076436102676480 |
|---|---|
| author | Derakhshan, J Macintyre, A |
| author_facet | Derakhshan, J Macintyre, A |
| author_sort | Derakhshan, J |
| collection | OXFORD |
| description | We prove that the theory of a Henselian valued field of characteristic zero, with finite ramification, and whose value group is a $Z$-group, is model-complete in the language of rings if the theory of its residue field is model-complete in the language of rings. We apply this to prove that every infinite algebraic extension of the field of $p$-adic numbers $\Bbb Q_p$ with finite ramification is model-complete in the language of rings. For this, we give a necessary and sufficient condition for model-completeness of the theory of a perfect pseudo-algebraically closed field with pro-cyclic absolute Galois group. |
| first_indexed | 2024-03-07T00:03:44Z |
| format | Journal article |
| id | oxford-uuid:76d8631a-7f41-4643-965e-3c394721b29d |
| institution | University of Oxford |
| last_indexed | 2024-03-07T00:03:44Z |
| publishDate | 2016 |
| publisher | Cornell University |
| record_format | dspace |
| spelling | oxford-uuid:76d8631a-7f41-4643-965e-3c394721b29d2022-03-26T20:19:02ZModel completeness for Henselian fields with finite ramification valued in a $Z$-groupJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:76d8631a-7f41-4643-965e-3c394721b29dSymplectic Elements at OxfordCornell University2016Derakhshan, JMacintyre, AWe prove that the theory of a Henselian valued field of characteristic zero, with finite ramification, and whose value group is a $Z$-group, is model-complete in the language of rings if the theory of its residue field is model-complete in the language of rings. We apply this to prove that every infinite algebraic extension of the field of $p$-adic numbers $\Bbb Q_p$ with finite ramification is model-complete in the language of rings. For this, we give a necessary and sufficient condition for model-completeness of the theory of a perfect pseudo-algebraically closed field with pro-cyclic absolute Galois group. |
| spellingShingle | Derakhshan, J Macintyre, A Model completeness for Henselian fields with finite ramification valued in a $Z$-group |
| title | Model completeness for Henselian fields with finite ramification valued in a $Z$-group |
| title_full | Model completeness for Henselian fields with finite ramification valued in a $Z$-group |
| title_fullStr | Model completeness for Henselian fields with finite ramification valued in a $Z$-group |
| title_full_unstemmed | Model completeness for Henselian fields with finite ramification valued in a $Z$-group |
| title_short | Model completeness for Henselian fields with finite ramification valued in a $Z$-group |
| title_sort | model completeness for henselian fields with finite ramification valued in a z group |
| work_keys_str_mv | AT derakhshanj modelcompletenessforhenselianfieldswithfiniteramificationvaluedinazgroup AT macintyrea modelcompletenessforhenselianfieldswithfiniteramificationvaluedinazgroup |