On the Galois group of Generalized Laguerre Polynomials

Details On the Galois group of Generalized Laguerre Polynomials On the Galois group of Generalized Laguerre Polynomials

2004-06-15

arxiv.org/abs/math/0406308v1

Mathematics

Number Theory

6 pages

Scientific paper

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 $\alpha=0,1$.

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