  For a suitable irreducible \textit{base} polynomial $f(x)\in \mathbf{Z}[x]$
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 $(\Phi_{d,f}(x))=\varphi(d)\cdot\mathrm{deg} (f)$ 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
  $S_3 \oplus C_2 , S_3 \oplus C_2\oplus C_2$
  and $S_3 \oplus C_4$ respectively.
