On Galois groups of prime degree polynomials with complex roots
Let \(f\) be an irreducible polynomial of prime degree \(p\geq 5\) over \({\mathbb Q}\), with precisely \(k\) pairs of complex roots. Using a result of Jens Hochsmann (1999), show that if \(p\geq 4k+1\) then \(\operatorname{Gal}(f/{\mathbb Q})\) is isomorphic to \(A_{p}\) or \(S_{p}\). This improv...
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/780 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-7802018-04-04T08:38:30Z On Galois groups of prime degree polynomials with complex roots Ben-Shimol, Oz 20B35; 12F12 Let \(f\) be an irreducible polynomial of prime degree \(p\geq 5\) over \({\mathbb Q}\), with precisely \(k\) pairs of complex roots. Using a result of Jens Hochsmann (1999), show that if \(p\geq 4k+1\) then \(\operatorname{Gal}(f/{\mathbb Q})\) is isomorphic to \(A_{p}\) or \(S_{p}\). This improves the algorithm for computing the Galois group of an irreducible polynomial of prime degree, introduced by A. Bialostocki and T. Shaska.If such a polynomial \(f\) is solvable by radicals then its Galois group is a Frobenius group of degree p. Conversely, any Frobenius group of degree p and of even order, can be realized as the Galois group of an irreducible polynomial of degree \(p\) over \({\mathbb Q}\) having complex roots. 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/780 Algebra and Discrete Mathematics; Vol 8, No 2 (2009) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/780/310 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
20B35; 12F12 |
| spellingShingle |
20B35; 12F12 Ben-Shimol, Oz On Galois groups of prime degree polynomials with complex roots |
| topic_facet |
20B35; 12F12 |
| format |
Article |
| author |
Ben-Shimol, Oz |
| author_facet |
Ben-Shimol, Oz |
| author_sort |
Ben-Shimol, Oz |
| title |
On Galois groups of prime degree polynomials with complex roots |
| title_short |
On Galois groups of prime degree polynomials with complex roots |
| title_full |
On Galois groups of prime degree polynomials with complex roots |
| title_fullStr |
On Galois groups of prime degree polynomials with complex roots |
| title_full_unstemmed |
On Galois groups of prime degree polynomials with complex roots |
| title_sort |
on galois groups of prime degree polynomials with complex roots |
| description |
Let \(f\) be an irreducible polynomial of prime degree \(p\geq 5\) over \({\mathbb Q}\), with precisely \(k\) pairs of complex roots. Using a result of Jens Hochsmann (1999), show that if \(p\geq 4k+1\) then \(\operatorname{Gal}(f/{\mathbb Q})\) is isomorphic to \(A_{p}\) or \(S_{p}\). This improves the algorithm for computing the Galois group of an irreducible polynomial of prime degree, introduced by A. Bialostocki and T. Shaska.If such a polynomial \(f\) is solvable by radicals then its Galois group is a Frobenius group of degree p. Conversely, any Frobenius group of degree p and of even order, can be realized as the Galois group of an irreducible polynomial of degree \(p\) over \({\mathbb Q}\) having complex roots. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/780 |
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AT benshimoloz ongaloisgroupsofprimedegreepolynomialswithcomplexroots |
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2024-04-12T06:26:17Z |
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2024-04-12T06:26:17Z |
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1796109162944397312 |