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We show that the discriminant of the generalized Laguerre polynomial L n (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 L n (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 L (α) n (x) over Q is An is α = n.

On classifying Laguerre polynomials which have Galois group the alternating group