On the Galois group of Generalized Laguerre Polynomials

Abstract

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be ``large.'' For a fixed $α\in \Q - \Z_{<0}$, Filaseta and Lam have shown that the $n$th degree Generalized Laguerre Polynomial $L_n^{(\alpha )}(x)={\sum }_{j=0}^n\left(\genfrac{}{}{0ex}{}{n+\alpha }{n-j}\right)(-x{)}^j/j!$ 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 $\alpha =0,1$.

6 pages

References in corpus (1)

• Algebraic properties of a family of Generalized Laguerre Polynomials

Cited by in corpus (1)

• Algebraic properties of a family of Generalized Laguerre Polynomials