On the Galois Group of generalized Laguerre polynomials
نویسنده
چکیده
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<0, Filaseta and Lam have shown that the nth degree Generalized Laguerre Polynomial L (α) n (x) = ∑n j=0 ( n+α n−j ) (−x)/j! is irreducible for all large enough n. We use our criterion to show that, under these conditions, the Galois group of L (α) n (x) is either the alternating or symmetric group on n letters, generalizing results of Schur for α = 0, 1.
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