We show that the discriminant of the generalized Laguerre polynomial is a non-zero square for some integer pair , with , if and only if belongs to one of 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 over is the alternating group . For example, we establish that for all but finitely many positive integers , the only for which the Galois group of over is is .
Nous démontrons que le discriminant du polynôme de Laguerre généralisé , pour un couple d’entiers avec , n’est le carré d’un entier non nul que si fait partie d’une trentaine d’ensembles explicites et infinis ou si fait partie d’un ensemble supplémentaire qui est fini. Donc nous obtenons de nouvelles informations concernant la réalisation du groupe alterné comme groupe de Galois du polynôme sur les nombres rationnels . Par exemple, nous établissons que pour tous les entiers positifs avec (avec un nombre fini de cas exceptionnels), la seule valeur d’ pour laquelle le groupe de Galois est le groupe alterné est le cas où .
Mots-clés : Generalized Laguerre polynomials, discriminants
Pradipto Banerjee 1; Michael Filaseta 2; Carrie E. Finch 3; J. Russell Leidy 2
@article{JTNB_2013__25_1_1_0, 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}, doi = {10.5802/jtnb.822}, mrnumber = {3063827}, zbl = {1287.11128}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.822/} }
TY - JOUR AU - Pradipto Banerjee AU - Michael Filaseta AU - Carrie E. Finch AU - J. Russell Leidy TI - On classifying Laguerre polynomials which have Galois group the alternating group JO - Journal de théorie des nombres de Bordeaux PY - 2013 SP - 1 EP - 30 VL - 25 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.822/ DO - 10.5802/jtnb.822 LA - en ID - JTNB_2013__25_1_1_0 ER -
%0 Journal Article %A Pradipto Banerjee %A Michael Filaseta %A Carrie E. Finch %A J. Russell Leidy %T On classifying Laguerre polynomials which have Galois group the alternating group %J Journal de théorie des nombres de Bordeaux %D 2013 %P 1-30 %V 25 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.822/ %R 10.5802/jtnb.822 %G en %F JTNB_2013__25_1_1_0
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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