Let \(n\geq 1\) and \(\zeta_n \coloneqq e^{2\pi i/n}\in\CC\). The \(n\)th cyclotomic polynomial is the product over the primitive \(n\)th roots of unity \[ \Phi_{n}(x) \coloneqq \prod_{\substack{1\leq k\leq n \\ \gcd(k, n)=1}}\left(x- \zeta_n^k\right) \in \CC[x]. \]
Let \(n\geq 1\). Then \(\Phi_n(x)\) lies in \(\ZZ[x]\), is monic of degree \(\phi(n)\), where \(\phi\) is Euler's totient function, and is irreducible in \(\QQ[x]\). Moreover, \(\Phi_n\) is the unique monic irreducible polynomial in \(\QQ[x]\) that divides \(x^n - 1\) and does not divide \(x^k-1\) for any \(1\leq k<n\).