An infinite, two-parameter family of polynomials with factorization similar to Xm-1

For a suitable irreducible base polynomial f(x) is an element of Z[x] of degree k, a family of polynomials F-m(x) depending on f(x) is constructed with the properties: (i) there is exactly one irreducible factor Phi(d,f)(x) for F-m(x) for each divisor d of m; (ii) deg(Phi(d,f)(x)) = phi(d) center dot deg(f) generalizing the factorization of x(m) - 1 into cyclotomic polynomials; (iii) when the base polynomial f(x) = x - 1 this F-m(x) coincides with x(m) - 1. As an application, irreducible polynomials of degree 12, 24, 24 are constructed having Galois groups of order matching their degrees and isomorphic to S-3 circle plus C-2, S-3 circle plus C-2 circle plus C-2 and S-3 circle plus C-4, respectively.