Cyclotomic factorization
= Cyclotomic factorization
Partitioning the roots of $X^n-1$ by their exact orders gives
$$
X^n-1=\prod_{d\mid n}\Phi_d(X).
$$
Inductively, 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 $\Phi_n\in\mathbb Z[X]$.