On the irreducibility of a certain class of Laguerre polynomials

R. Gow has investigated the problem of determining classical polynomials with Galois group A(m), the alternating group on tit letters, in the case that in is even (odd m being previously handled in work of I. Schur). He showed that the generalized Laguerre polynomial L-m((m))(x), defined below, has Galois group A(m) provided m > 2 is even and L-m((m)) (x) is irreducible (and obtained irreducibility in some cases). In this paper, we establish that L-m((m))(x) is irreducible for almost all m (and, hence, has Galois group A(m) for almost all even m). (C) 2003 Elsevier Science (USA). All rights reserved.