Defining Transcendentals in Function Fields.
Given any field K, there is a function field F/K in one variable containing definable transcendentals over K, i,e., elements in F/K first-order definable in the language of fields with parameters from K. Hence, the model-theoretic and the field-theoretic relative algebraic closure of K in F do not c...
| Үндсэн зохиолч: | |
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| Формат: | Journal article |
| Хэл сонгох: | English |
| Хэвлэсэн: |
2002
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| _version_ | 1826257374621466624 |
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| author | Koenigsmann, J |
| author_facet | Koenigsmann, J |
| author_sort | Koenigsmann, J |
| collection | OXFORD |
| description | Given any field K, there is a function field F/K in one variable containing definable transcendentals over K, i,e., elements in F/K first-order definable in the language of fields with parameters from K. Hence, the model-theoretic and the field-theoretic relative algebraic closure of K in F do not coincide. E.g., if K is finite, the model-theoretic algebraic closure of K in the rational function field K(t) is K(t). For the proof, diophantine ∅-definability of K in F is established for any function field F/K in one variable, provided K is large, or K x /(K x)n is finite for some integer n > 1 coprime to char K. |
| first_indexed | 2024-03-06T18:17:12Z |
| format | Journal article |
| id | oxford-uuid:05037800-7abc-4240-8256-29047fe48c77 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T18:17:12Z |
| publishDate | 2002 |
| record_format | dspace |
| spelling | oxford-uuid:05037800-7abc-4240-8256-29047fe48c772022-03-26T08:54:48ZDefining Transcendentals in Function Fields.Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:05037800-7abc-4240-8256-29047fe48c77EnglishSymplectic Elements at Oxford2002Koenigsmann, JGiven any field K, there is a function field F/K in one variable containing definable transcendentals over K, i,e., elements in F/K first-order definable in the language of fields with parameters from K. Hence, the model-theoretic and the field-theoretic relative algebraic closure of K in F do not coincide. E.g., if K is finite, the model-theoretic algebraic closure of K in the rational function field K(t) is K(t). For the proof, diophantine ∅-definability of K in F is established for any function field F/K in one variable, provided K is large, or K x /(K x)n is finite for some integer n > 1 coprime to char K. |
| spellingShingle | Koenigsmann, J Defining Transcendentals in Function Fields. |
| title | Defining Transcendentals in Function Fields. |
| title_full | Defining Transcendentals in Function Fields. |
| title_fullStr | Defining Transcendentals in Function Fields. |
| title_full_unstemmed | Defining Transcendentals in Function Fields. |
| title_short | Defining Transcendentals in Function Fields. |
| title_sort | defining transcendentals in function fields |
| work_keys_str_mv | AT koenigsmannj definingtranscendentalsinfunctionfields |