On the Galois group of generalized Laguerre polynomials

Hajir, Farshid

doi:10.5802/jtnb.505

Hajir, Farshid ¹

¹ Department of Mathematics & Statistics University of Massachusetts Amherst, MA 01003-9318 USA

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 517-525.

Résumé
Abstract

En utilisant la théorie des polygones de Newton, on obtient un critère simple pour montrer que le groupe de Galois d’un polynôme soit “grand.” Si on fixe $α \in ℚ - ℤ_{< 0}$ , Filaseta et Lam ont montré que le Polynôme Generalisé de Laguerre $L_{n}^{(α)} (x) = \sum_{j = 0}^{n} (\binom{n + α}{n - j}) {(- x)}^{j} / j!$ est irréductible quand le degré $n$ est assez grand. On utilise notre critère afin de montrer que, sous ces hypothèses, le groupe de Galois de $L_{n}^{(α)} (x)$ est soit le groupe alterné, soit le groupe symétrique, de degré $n$ , généralisant des résultats de Schur pour $α = 0, 1, \pm \frac{1}{2}, - 1 - n$ .

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed $α \in ℚ - ℤ_{< 0}$ , Filaseta and Lam have shown that the $n$ th degree Generalized Laguerre Polynomial $L_{n}^{(α)} (x) = \sum_{j = 0}^{n} (\binom{n + α}{n - j}) {(- x)}^{j} / j!$ is irreducible for all large enough $n$ . We use our criterion to show that, under these conditions, the Galois group of $L_{n}^{(α)} (x)$ is either the alternating or symmetric group on $n$ letters, generalizing results of Schur for $α = 0, 1, \pm \frac{1}{2}, - 1 - n$ .

MR Zbl | 2 citations dans Numdam

DOI : 10.5802/jtnb.505

Mots-clés : Galois group, Generalized Laguerre Polynomial, Newton Polygon

Affiliations des auteurs :

Hajir, Farshid ¹

¹ Department of Mathematics & Statistics University of Massachusetts Amherst, MA 01003-9318 USA

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Hajir, Farshid. On the Galois group of generalized Laguerre polynomials. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 517-525. doi : 10.5802/jtnb.505. https://www.numdam.org/articles/10.5802/jtnb.505/

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