  Using the theory of Newton Polygons, we formulate a simple criterion for the
Galois group of a polynomial to be ``large.'' For a fixed $\alpha \in \Q -
\Z_{<0}$, Filaseta and Lam have shown that the $n$th degree Generalized
Laguerre Polynomial $L_n^{(\alpha)}(x) = \sum_{j=0}^n
\binom{n+\alpha}{n-j}(-x)^j/j!$ is irreducible for all large enough $n$. We use
our criterion to show that, under these conditions, the Galois group of $\La$
is either the alternating or symmetric group on $n$ letters, generalizing
results of Schur for $\alpha=0,1$.
