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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 $f(x)$ isconstructed with the properties:  (i) there is exactly one irreducible factor $\Phi_{d,f}(x)$ for $F_m(x)$ foreach divisor $d$ of $m$;  (ii) deg $(\Phi_{d,f}(x))=\varphi(d)\cdot\mathrm{deg} (f)$ generalizing thefactorization 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 areconstructed having Galois groups of order matching their degrees and isomorphicto  $S_3 \oplus C_2 , S_3 \oplus C_2\oplus C_2$  and $S_3 \oplus C_4$ respectively.

An Infinite, Two-parameter Family of Polynomials with Factorization Similar to $X^m-1$