We show that the discriminant of the generalized Laguerre polynomial Ln(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 Ln(α) (x) over Q is the alternating group An. 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 Ln(α) (x) over Q is An is α = n. © Société Arithmétique de Bordeaux, 2013, tous droits réservés.