On the irreducibility of a certain class of Laguerre polynomials Original Research Article

R. Gow has investigated the problem of determining classical polynomials with Galois group Am, the alternating group on m letters, in the case that m is even (odd m being previously handled in work of I. Schur). He showed that the generalized Laguerre polynomial Lm(m)(x), defined below, has Galois group Am provided m>2 is even and Lm(m)(x) is irreducible (and obtained irreducibility in some cases). In this paper, we establish that Lm(m)(x) is irreducible for almost all m (and, hence, has Galois group Am for almost all even m).