LAGUERRE POLYNOMIALS WITH GALOIS GROUP Am FOR EACH

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 Sm, the symmetric group on m letters, and there exists a polynomial of degree m with rational coefficients and associated Galois group Am, the alternating group on m letters. In the late 1920’s and early 1930’s, I. Schur found concrete examples of such polynomials among the classical Laguerre polynomials except in the case of polynomials with Galois group Am 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 Am.