Some generalized Laguerre polynomials whose Galois groups are the alternating groups
type: Item
instance of: scholarly article
title: Some generalized Laguerre polynomials whose Galois groups are the alternating groups
zbMATH Open document ID: 0693.12009
DOI: 10.1016/0022-314X(89)90021-8
author: Jianyong Qiao
author: Sumit K. Garg
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: The author considers one kind of generalized Laguerre polynomials \[ F_n = (-1)^n n!\sum^n_{m=0}\binom{2n}{n-m}\frac{(-X)^m}{m!}\in\mathbb Z[X] \] and shows that their Galois groups over \(\mathbb Q\) are the alternating group \(A_n\) if \(F_n\) is irreducible over \(\mathbb Q\) and \(n\) is even. He is able to prove that \(F_n\) is irreducible over \(\mathbb Q\) if \(n=2p^k\), \(p\ge 5\) a prime or \(n=4p^k\), \(p\ge 11\) a prime. These results are almost in the sense of I. Schur who gave the first families of rational polynomials whose Galois groups are \(A_n\) in cases \(n\equiv 0\pmod 4\) or \(n\equiv 1\pmod 2\).
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 4134172
zbMATH Keywords: inverse Galois problem
zbMATH Keywords: generalized Laguerre polynomials
zbMATH Keywords: alternating group
reviewed by: Bernd Richter
MaRDI profile type: Publication
cites work: Addendum to "On Some Algebraic Properties of the Bessel Polynomials"
description: scientific article; zbMATH DE number 4134172