Galois groups of polynomials arising from circulant matrices

Abstract

Computing the Galois group of the splitting field of a given polynomial with integer coeffi- cients 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 S(n), the symmetric group on n letters. Thus, cases where the associated Galois group is different from S(n) are rare. Nevertheless, examples of polynomials where the associated Galois group is not S(n) are well-known.