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The cyclotomic polynomial iswhere ${\zeta }_n$ is primitive. Partitioning all $n$th roots by exact order gives the cyclotomic factorizationInduct on $n$. The quotient of the monic integer polynomial $X^n-1$ by the product of the already constructed monic ${\Phi }_d$, $d<n$, lies in $\mathbb{Q}[X]$ and is monic. Gauss lemma then shows that this quotient ${\Phi }_n$ lies in $\mathbb{Z}[X]$.

If the prime $p$ does not divide $n$, then over ${\mathbb{F}}_p$,Thus $X^n-1$ has no repeated root. Every factor, including the reduction of ${\Phi }_n$, is square-free, proving the Separability of a cyclotomic polynomial modulo p.