In this paper, we compute Galois groups over the rationals associated with generalized Laguerre polynomials Ln(α)(x) whose discriminants are rational squares, where n and α are integers. An explicit description of the integer pairs (n, α) for which the discriminant of Ln(α)(x) is a rational square was recently obtained by the author in a joint work with Filaseta, Finch and Leidy. Among these pairs (n, α), we show that for 2 ≤ n ≤ 5, the associated Galois group of Ln(α)(x) is always A n, 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. © 2014 Elsevier Inc.