On a certain family of generalized Laguerre polynomials

Following the work of Schur and Coleman, we prove the generalized Laguerre polynomial is irreducible over the rationals for all n?1 and has Galois group A_n if n+1 is an odd square, and S_n otherwise. We also show that for certain negative integer values of α and certain congruence classes of n modulo 8, the splitting field of L_n^(α)(x) can be embedded in a double cover.