\(2\)-Galois groups and the Kaplansky radical
An accurate description of the Galois group \(G_{F}(2)\) of the maximal Galois \(2\)-extension of a field \(F\) may be given for fields \(F\) admitting a \(2\)-henselian valuation ring. In this note we generalize this result by characterizing the fields for which \({G_{F}{(2)}}\) decomposes as a fre...
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| Date: | 2018 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2018
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/649 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543311899131904 |
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| author | Dario, Ronie Peterson Engler, Antonio Jose |
| author_facet | Dario, Ronie Peterson Engler, Antonio Jose |
| author_sort | Dario, Ronie Peterson |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-04-04T09:17:05Z |
| description | An accurate description of the Galois group \(G_{F}(2)\) of the maximal Galois \(2\)-extension of a field \(F\) may be given for fields \(F\) admitting a \(2\)-henselian valuation ring. In this note we generalize this result by characterizing the fields for which \({G_{F}{(2)}}\) decomposes as a free pro-\(2\) product \(\mathcal{F}*\mathcal{H}\) where \(\mathcal{F}\) is a free closed subgroup of \({G_{F}{(2)}}\) and \(\mathcal{H}\) is the Galois group of a \(2\)-henselian extension of \(F\). The free product decomposition of \({G_{F}{(2)}}\) is equivalent to the existence of a valuation ring compatible with the Kaplansky radical of \(F\). Fields with Kaplansky radical fulfilling prescribed conditions are constructed, as an application. |
| first_indexed | 2025-12-02T15:27:11Z |
| format | Article |
| id | admjournalluguniveduua-article-649 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:27:11Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-6492018-04-04T09:17:05Z \(2\)-Galois groups and the Kaplansky radical Dario, Ronie Peterson Engler, Antonio Jose Brauer group, free pro-\(2\) product, Galois group, \(2\)-henselian valuation ring, quadratic form 12J10; 12F10 An accurate description of the Galois group \(G_{F}(2)\) of the maximal Galois \(2\)-extension of a field \(F\) may be given for fields \(F\) admitting a \(2\)-henselian valuation ring. In this note we generalize this result by characterizing the fields for which \({G_{F}{(2)}}\) decomposes as a free pro-\(2\) product \(\mathcal{F}*\mathcal{H}\) where \(\mathcal{F}\) is a free closed subgroup of \({G_{F}{(2)}}\) and \(\mathcal{H}\) is the Galois group of a \(2\)-henselian extension of \(F\). The free product decomposition of \({G_{F}{(2)}}\) is equivalent to the existence of a valuation ring compatible with the Kaplansky radical of \(F\). Fields with Kaplansky radical fulfilling prescribed conditions are constructed, as an application. Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/649 Algebra and Discrete Mathematics; Vol 10, No 2 (2010) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/649/183 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | Brauer group free pro-\(2\) product Galois group \(2\)-henselian valuation ring quadratic form 12J10 12F10 Dario, Ronie Peterson Engler, Antonio Jose \(2\)-Galois groups and the Kaplansky radical |
| title | \(2\)-Galois groups and the Kaplansky radical |
| title_full | \(2\)-Galois groups and the Kaplansky radical |
| title_fullStr | \(2\)-Galois groups and the Kaplansky radical |
| title_full_unstemmed | \(2\)-Galois groups and the Kaplansky radical |
| title_short | \(2\)-Galois groups and the Kaplansky radical |
| title_sort | \(2\)-galois groups and the kaplansky radical |
| topic | Brauer group free pro-\(2\) product Galois group \(2\)-henselian valuation ring quadratic form 12J10 12F10 |
| topic_facet | Brauer group free pro-\(2\) product Galois group \(2\)-henselian valuation ring quadratic form 12J10 12F10 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/649 |
| work_keys_str_mv | AT darioroniepeterson 2galoisgroupsandthekaplanskyradical AT englerantoniojose 2galoisgroupsandthekaplanskyradical |