Cyclotomic factorization

Partitioning the roots of $X^n-1$ by their exact orders givesInductively, the product over proper divisors is a monic polynomial in $\mathbb{Z}[X]$. Division of $X^n-1$ by this monic polynomial stays in $\mathbb{Z}[X]$ and has zero remainder, proving that ${\Phi }_n\in \mathbb{Z}[X]$.