On Galois groups of generalized Laguerre polynomials whose discriminants are squares
type: Item
instance of: scholarly article
title: On Galois groups of generalized Laguerre polynomials whose discriminants are squares
zbMATH Open document ID: 1322.11116
author: Pradipto Banerjee
published in: Journal of Number Theory
publication date: 2014-08-28T00:00:00Z
reviewed by: Teresa Crespo
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R09
zbMATH DE Number: 6335258
zbMATH Keywords: generalized Laguerre polynomials
zbMATH Keywords: irreducibility of polynomials
zbMATH Keywords: Galois group
describes a project that uses: SageMath
MaRDI profile type: Publication
full work available at URL: j.jnt.2014.01.009
OpenAlex ID: W2089560761
cites work: On classifying Laguerre polynomials which have Galois group the alternating group
cites work: THE DIOPHANTINE EQUATIONS x3=Ny2±1
cites work: Laguerre polynomials with Galois group \(A_m\) for each \(m\)
cites work: On the irreducibility of the generalized Laguerre polynomials
cites work: Some generalized Laguerre polynomials whose Galois groups are the alternating groups
cites work: Some \(\widetilde A_ n\)-extensions obtained from generalized Laguerre polynomials
cites work: On the Galois group of generalized Laguerre polynomials
cites work: Algebraic Properties of a Family of Generalized Laguerre Polynomials
cites work: Specializations of one-parameter families of polynomials
cites work: An Elementary Test for the Galois Group of a Quartic Polynomial
cites work: Approximate formulas for some functions of prime numbers
cites work: Q5181335
cites work: On a certain family of generalized Laguerre polynomials
cites work: Class numbers of imaginary quadratic fields
DOI: 10.1016/J.JNT.2014.01.009
review text: The author computes the Galois group of Laguerre polynomials NEWLINE\[NEWLINEL_n^{(\alpha)}(x)=\sum_{j=0}^n \dfrac{(n+\alpha)(n-1+\alpha) \cdots (j+1+\alpha)}{(n-j)!j!} (-x)^j,NEWLINE\]NEWLINE with \(n\) and \(\alpha\) integer numbers, whose discriminant is a square. In [the author et al., J. Théor. Nombres Bordx. 25, No. 1, 1--30 (2013; Zbl 1287.11128)], an explicit description of the integer pairs \((n,\alpha)\) for which the discriminant of \(L_n^{(\alpha)}(x)\) is a rational square was obtained.NEWLINENEWLINEIn the paper under review, it is shown that, among these pairs, for \(2\leq n \leq 5\), the Galois group of \(L_n^{(\alpha)}(x)\) is the alternating group \(A_n\) except for the pairs \((4,-1)\) and \((4,23)\). For \(n\geq 6\), it is shown that the Galois group of \(L_n^{(\alpha)}(x)\) is \(A_n\) if and only if \(L_n^{(\alpha)}(x)\) is irreducible over the field of rational numbers. The proof uses the Newton polygons of the \(L_n^{(\alpha)}(x)\) as well as results of \textit{G. Dumas} [Journ. de Math. (6) 2, 191--258 (1906; JFM 37.0096.01)] to establish irreducibility, and of \textit{F. Hajir} [J. Théor. Nombres Bordx. 17, No. 2, 517--525 (2005; Zbl 1094.11042)] to determine the Galois groups. For the pairs \((n,\alpha)\) for which the discriminant of \(L_n^{(\alpha)}(x)\) is a rational square and \(n\leq 35\), the author has checked, using Sage, that the Galois group of \(L_n^{(\alpha)}(x)\) is \(A_n\), either directly or checking that \(L_n^{(\alpha)}(x)\) is irreducible.
description: scientific article; zbMATH DE number 6335258