Laguerre polynomials with Galois groupAmfor each m

Abstract

In 1892, D. Hilbert began what is now called Inverse Galois Theory by showing that for each positive integer m , there exists a polynomial of degree m with rational coefficients and associated Galois group S m , the symmetric group on m letters, and there exists a polynomial of degree m with rational coefficients and associated Galois group A m , the alternating group on m letters. In the late 1920s and early 1930s, I. Schur found concrete examples of such polynomials among the classical Laguerre polynomials except in the case of polynomials with Galois group A m where m ≡ 2 ( mod 4 ) . Following up on work of R. Gow from 1989, this paper complements the work of Schur by showing that for every positive integer m ≡ 2 ( mod 4 ) , there is in fact a Laguerre polynomial of degree m with associated Galois group A m .