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Algebra and Discrete Mathematics, 2009, Issue 2, Pages 99–107
(Mi adm122)
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This article is cited in 2 scientific papers (total in 2 papers)
RESEARCH ARTICLE
On Galois groups of prime degree polynomials with complex roots
Oz Ben-Shimol Department of Mathematics, University of Haifa, Mount Carmel 1905, Haifa, Israel
Abstract:
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 Höchsmann (1999), we show that if $p\geq 4k+1$ then $\rm{Gal}(f/\mathbb{Q})$ 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.
Received: 06.09.2008 Revised: 15.04.2009
Citation:
Oz Ben-Shimol, “On Galois groups of prime degree polynomials with complex roots”, Algebra Discrete Math., 2009, no. 2, 99–107
Linking options:
https://www.mathnet.ru/eng/adm122 https://www.mathnet.ru/eng/adm/y2009/i2/p99
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Abstract page: | 332 | Full-text PDF : | 278 | First page: | 1 |
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