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 ∈ Q − Z<0, Filaseta and Lam have shown that the nth degree Generalized Laguerre Polynomial L () n (x) = P n=0 n+ n j (−x) j /j! is irreducible for all large enough n. We use our criterion to show that, under these condi- tions, the Galois group of L () n (x) is either the alternating or symmetric group on n letters, generalizing results of Schur for = 0,1.