Galois Groups of Polynomials Arising from Circulant Matrices

Computing the Galois group of the splitting field of a given polynomial with integer coefficients 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 x − 1, p ≥ 3 prime, is cyclic of order p− 1. For the polynomial x − 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. An interest in this paper is to find other collections of polynomials with integer coefficients whose Galois groups are isomorphic to these groups. Using circulant matrices, we are led in the next section to the polynomials