On Galois groups of generalized Laguerre polynomials whose discriminants are squares

JOURNAL OF NUMBER THEORY

卷:141

Article

Banerjee, Pradipto

关键词: Generalized Laguerre polynomials;    Irreducibility of polynomials;    Galois groups of cubics, quartics and quintics;   

DOI  :  10.1016/j.jnt.2014.01.009

来源: Elsevier

【 摘 要 】

In this paper, we compute Galois groups over the rationals associated with generalized Laguerre polynomials L-n((alpha)) (x) whose discriminants are rational squares, where n and alpha are integers. An explicit description of the integer pairs (n, alpha) for which the discriminant of L-n((alpha)) (x) is a rational square was recently obtained by the author in a joint work with Filaseta, Finch and Leidy. Among these pairs (n, alpha), we show that for 2 <= n <= 5, the associated Galois group of L-n((alpha)) (x) is always An, except for the pairs (4, -1) and (4,23). For n >= 6, we show that the corresponding Galois group is A(n) if and only if the polynomial concerned is irreducible over the rationals. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free