On the Galois Group of generalized Laguerre polynomials

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed α∈ℚ-ℤ \u3c0 , Filaseta and Lam have shown that the nth degree Generalized Laguerre Polynomial L n (α) (x)=∑ j=0 n n+α n-j(-x) j /j! is irreducible for all large enough n. We use our criterion to show that, under these conditions, the Galois group of L n (α) (x) is either the alternating or symmetric group on n letters, generalizing results of Schur for α=0,1,±1 2,-1-n