= 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 $f=\Phi_n$.
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