\(2\)-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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Lugansk National Taras Shevchenko University
2018
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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 |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2018-04-04T09:17:05Z |
| collection |
OJS |
| language |
English |
| topic |
Brauer group free pro-\(2\) product Galois group \(2\)-henselian valuation ring quadratic form 12J10 12F10 |
| 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 |
| topic_facet |
Brauer group free pro-\(2\) product Galois group \(2\)-henselian valuation ring quadratic form 12J10 12F10 |
| format |
Article |
| author |
Dario, Ronie Peterson Engler, Antonio Jose |
| author_facet |
Dario, Ronie Peterson Engler, Antonio Jose |
| author_sort |
Dario, Ronie Peterson |
| title |
\(2\)-Galois groups and the Kaplansky radical |
| title_short |
\(2\)-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_sort |
\(2\)-galois groups and the kaplansky radical |
| 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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/649 |
| work_keys_str_mv |
AT darioroniepeterson 2galoisgroupsandthekaplanskyradical AT englerantoniojose 2galoisgroupsandthekaplanskyradical |
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2025-12-02T15:27:11Z |
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2025-12-02T15:27:11Z |
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1850411879750434816 |