On Galois groups of generalized Laguerre polynomials whose discriminants are squares

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)(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)(n,α), we show that for 2⩽n⩽52⩽n⩽5, the associated Galois group of Ln(α)(x) is always AnAn, except for the pairs (4,−1)(4,−1) and (4,23)(4,23). For n⩾6n⩾6, we show that the corresponding Galois group is AnAn if and only if the polynomial concerned is irreducible over the rationals.