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 An if n+1 is an odd square, and Sn otherwise. We also show that for certain negative integer values of α and certain congruence classes of n modulo 8, the splitting field of Ln(α)(x) can be embedded in a double cover.