 Irreducibility of cyclotomic polynomials

Every cyclotomic polynomial ${\Phi }_n$ is an irreducible polynomial over $\mathbb{Q}$. If a monic irreducible factor $f\in \mathbb{Z}[X]$ contains a primitive root $\zeta$, then it also contains ${\zeta }^p$ for every prime number $p\nmid n$: otherwise, writing ${\Phi }_n=fg$, reduction modulo $p$ and the Frobenius endomorphism give $\bar{f}\mid \bar{g}(X{)}^p$. Some irreducible factor would then divide both $\bar{f}$ and $\bar{g}$, giving a repeated factor of $X^n-1$, contrary to separability of roots of unity. Iterating over the prime factors of every integer coprime to $n$ shows that $f$ contains all primitive $n$th roots, so $f={\Phi }_n$.