On classifying Laguerre polynomials which have Galois group the alternating group

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

Journal de Théorie des Nombres de Bordeaux (2013)

  • Volume: 25, Issue: 1, page 1-30
  • ISSN: 1246-7405

Abstract

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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 n ≥ 1 , 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 ℚ is the alternating group A n . For example, we establish that for all but finitely many positive integers n ≡ 2 ( mod 4 ) , the only α for which the Galois group of L n ( α ) ( x ) over ℚ is A n is α = n .

How to cite

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Banerjee, Pradipto, et al. "On classifying Laguerre polynomials which have Galois group the alternating group." Journal de Théorie des Nombres de Bordeaux 25.1 (2013): 1-30. <http://eudml.org/doc/275766>.

@article{Banerjee2013,
abstract = {We show that the discriminant of the generalized Laguerre polynomial $L_\{n\}^\{(\alpha )\}(x)$ is a non-zero square for some integer pair $(n,\alpha )$, with $n \ge 1$, if and only if $(n,\alpha )$ 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 $L_\{n\}^\{(\alpha )\}(x)$ over $\mathbb\{Q\}$ is the alternating group $A_\{n\}$. For example, we establish that for all but finitely many positive integers $n \equiv 2 \hspace\{4.44443pt\}(\@mod \; 4)$, the only $\alpha $ for which the Galois group of $L_\{n\}^\{(\alpha )\}(x)$ over $\mathbb\{Q\}$ is $A_\{n\}$ is $\alpha = n$.},
affiliation = {Indian Statistical Institute Stat-Math Unit 203 Barrackpore Trunk Road Kolkata 700108, India; University of South Carolina Department of Mathematics 1523 Greene Street Columbia, SC 29208, USA; Washington and Lee University Mathematics Department Robinson Hall Lexington VA 24450, USA; University of South Carolina Department of Mathematics 1523 Greene Street Columbia, SC 29208, USA},
author = {Banerjee, Pradipto, Filaseta, Michael, Finch, Carrie E., Leidy, J. Russell},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Generalized Laguerre polynomials; discriminants; generalized Laguerre polynomials},
language = {eng},
month = {4},
number = {1},
pages = {1-30},
publisher = {Société Arithmétique de Bordeaux},
title = {On classifying Laguerre polynomials which have Galois group the alternating group},
url = {http://eudml.org/doc/275766},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Banerjee, Pradipto
AU - Filaseta, Michael
AU - Finch, Carrie E.
AU - Leidy, J. Russell
TI - On classifying Laguerre polynomials which have Galois group the alternating group
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/4//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 1
SP - 1
EP - 30
AB - We show that the discriminant of the generalized Laguerre polynomial $L_{n}^{(\alpha )}(x)$ is a non-zero square for some integer pair $(n,\alpha )$, with $n \ge 1$, if and only if $(n,\alpha )$ 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 $L_{n}^{(\alpha )}(x)$ over $\mathbb{Q}$ is the alternating group $A_{n}$. For example, we establish that for all but finitely many positive integers $n \equiv 2 \hspace{4.44443pt}(\@mod \; 4)$, the only $\alpha $ for which the Galois group of $L_{n}^{(\alpha )}(x)$ over $\mathbb{Q}$ is $A_{n}$ is $\alpha = n$.
LA - eng
KW - Generalized Laguerre polynomials; discriminants; generalized Laguerre polynomials
UR - http://eudml.org/doc/275766
ER -

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