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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| Datum: | 2018 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Lugansk National Taras Shevchenko University
2018
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| Online Zugang: | 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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Algebra and Discrete Mathematics| _version_ | 1856543435482202112 |
|---|---|
| author | Ben-Shimol, Oz |
| author_facet | Ben-Shimol, Oz |
| author_sort | Ben-Shimol, Oz |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-04-04T08:38:30Z |
| 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. |
| first_indexed | 2025-12-02T15:43:21Z |
| format | Article |
| id | admjournalluguniveduua-article-780 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:43:21Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-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 |
| spellingShingle | 20B35 12F12 Ben-Shimol, Oz On Galois groups of prime degree polynomials with complex roots |
| title | 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_short | On Galois groups of prime degree polynomials with complex roots |
| title_sort | on galois groups of prime degree polynomials with complex roots |
| topic | 20B35 12F12 |
| topic_facet | 20B35 12F12 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/780 |
| work_keys_str_mv | AT benshimoloz ongaloisgroupsofprimedegreepolynomialswithcomplexroots |