On classifying Laguerre polynomials which have Galois group the alternating group
Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 1, pp. 1-30.

We show that the discriminant of the generalized Laguerre polynomial L n (α) (x) is a non-zero square for some integer pair (n,α), with n1, if and only if (n,α) belongs to one of 30 explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of L n (α) (x) over is the alternating group A n . For example, we establish that for all but finitely many positive integers n2(mod4), the only α for which the Galois group of L n (α) (x) over is A n is α=n.

Nous démontrons que le discriminant du polynôme de Laguerre généralisé L n (α) (x), pour un couple (n,α) d’entiers avec n1, n’est le carré d’un entier non nul que si (n,α) fait partie d’une trentaine d’ensembles explicites et infinis ou si (n,α) fait partie d’un ensemble supplémentaire qui est fini. Donc nous obtenons de nouvelles informations concernant la réalisation du groupe alterné A n comme groupe de Galois du polynôme L n α (x) sur les nombres rationnels . Par exemple, nous établissons que pour tous les entiers positifs n avec n2(mod4) (avec un nombre fini de cas exceptionnels), la seule valeur d’α pour laquelle le groupe de Galois est le groupe alterné A n est le cas où α=n.

DOI: 10.5802/jtnb.822
Classification: 11R32,  11C08,  33C45
Keywords: Generalized Laguerre polynomials, discriminants
Pradipto Banerjee 1; Michael Filaseta 2; Carrie E. Finch 3; J. Russell Leidy 2

1 Indian Statistical Institute Stat-Math Unit 203 Barrackpore Trunk Road Kolkata 700108, India
2 University of South Carolina Department of Mathematics 1523 Greene Street Columbia, SC 29208, USA
3 Washington and Lee University Mathematics Department Robinson Hall Lexington VA 24450, USA
     author = {Pradipto Banerjee and Michael Filaseta and Carrie E. Finch and J. Russell Leidy},
     title = {On classifying {Laguerre} polynomials which have {Galois} group the alternating group},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {1--30},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {25},
     number = {1},
     year = {2013},
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Pradipto Banerjee; Michael Filaseta; Carrie E. Finch; J. Russell Leidy. On classifying Laguerre polynomials which have Galois group the alternating group. Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 1, pp. 1-30. doi : 10.5802/jtnb.822. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.822/

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