An infinite, two-parameter family of polynomials with factorization similar to X m − 1

Abstract

For a suitable irreducible base polynomial of degree k, a family of polynomials depending on f(x) is constructed with the properties: there is exactly one irreducible factor for for each divisor d of m; generalizing the factorization of into cyclotomic polynomials; when the base polynomial this coincides with As an application, irreducible polynomials of degree 12, 24, 24 are constructed having Galois groups of order matching their degrees and isomorphic to and respectively.