A journey through Galois groups, irreducible polynomials and diophatine equations

Abstract

Computing the Galois group of the splitting field of a given polynomial with integer coefficients over the rationals is a classical problem in modern algebra. A theorem of Van der Waerden [Wae] asserts that almost all (monic) polynomials in Z[x] have associated Galois group Sn, the symmetric group on n letters. Thus, cases where the associated Galois group is different from Sn are rare. Nevertheless, examples of polynomials where the associated Galois group is not Sn are well-known. For example, the Galois group of the splitting field of the polynomial xp − 1, p 3 prime, is cyclic of order p − 1. For the polynomial xp − 2, p > 3, the Galois group is the subgroup of Sp generated by a cycle of length p and a cycle of length p − 1. One interest in this paper is to find other collections of polynomials with integer or rational coefficients whose Galois groups are isomorphic to these groups.