Projective extensions of fields
Every field K admits proper projective extensions, that is, Galois extensions where the Galois group is a non-trivial projective group, unless K is separably closed or K is a Pythagorean formally real field without cyclic extensions of odd degree. As a consequence, it turns out that almost all absol...
| Main Author: | |
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| Format: | Journal article |
| Language: | English |
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2006
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| _version_ | 1797077264115957760 |
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| author | Koenigsmann, J |
| author_facet | Koenigsmann, J |
| author_sort | Koenigsmann, J |
| collection | OXFORD |
| description | Every field K admits proper projective extensions, that is, Galois extensions where the Galois group is a non-trivial projective group, unless K is separably closed or K is a Pythagorean formally real field without cyclic extensions of odd degree. As a consequence, it turns out that almost all absolute Galois groups decompose as proper semidirect products. We show that each local field has a unique maximal projective extension, and that the same holds for each global field of positive characteristic. In characteristic 0, we prove that Leopoldt's conjecture for all totally real number fields is equivalent to the statement that, for all totally real number fields, all projective extensions are cyclotomic. So the realizability of any non-procyclic projective group as Galois group over ℚ produces counterexamples to the Leopoldt conjecture. © 2006 London Mathematical Society. |
| first_indexed | 2024-03-07T00:15:26Z |
| format | Journal article |
| id | oxford-uuid:7aadc8d0-5f78-4d3e-ab22-11e93519cb9f |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T00:15:26Z |
| publishDate | 2006 |
| record_format | dspace |
| spelling | oxford-uuid:7aadc8d0-5f78-4d3e-ab22-11e93519cb9f2022-03-26T20:45:34ZProjective extensions of fieldsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:7aadc8d0-5f78-4d3e-ab22-11e93519cb9fEnglishSymplectic Elements at Oxford2006Koenigsmann, JEvery field K admits proper projective extensions, that is, Galois extensions where the Galois group is a non-trivial projective group, unless K is separably closed or K is a Pythagorean formally real field without cyclic extensions of odd degree. As a consequence, it turns out that almost all absolute Galois groups decompose as proper semidirect products. We show that each local field has a unique maximal projective extension, and that the same holds for each global field of positive characteristic. In characteristic 0, we prove that Leopoldt's conjecture for all totally real number fields is equivalent to the statement that, for all totally real number fields, all projective extensions are cyclotomic. So the realizability of any non-procyclic projective group as Galois group over ℚ produces counterexamples to the Leopoldt conjecture. © 2006 London Mathematical Society. |
| spellingShingle | Koenigsmann, J Projective extensions of fields |
| title | Projective extensions of fields |
| title_full | Projective extensions of fields |
| title_fullStr | Projective extensions of fields |
| title_full_unstemmed | Projective extensions of fields |
| title_short | Projective extensions of fields |
| title_sort | projective extensions of fields |
| work_keys_str_mv | AT koenigsmannj projectiveextensionsoffields |