An infinite, two-parameter family of polynomials with factorization similar to <i>X</i>^<i>m</i> − 1

For a suitable irreducible \textit{base} polynomial $f(x)\in \mathbf{Z}[x]$ of degree $k$, family polynomials $F_m(x)$ depending on $f(x)$ is constructed with the properties: (i) there exactly one factor $\Phi_{d,f}(x)$ for each divisor $d$ $m$; (ii) deg $(\Phi_{d,f}(x))=\varphi(d)\cdot\mathrm{deg} (f)$ generalizing factorization $x^m-1$ into cyclotomic polynomials; (iii) when base $f(x) = x-1$ this coincides $x^m-1$. As an application, 12, 24, 24 are having Galois groups order matching their degrees and isomorphic to $S_3 \oplus C_2 , S_3 C_2\oplus C_2$ and $S_3 C_4$ respectively.