What is cyclotomic polynomial?

Let $n \geq 1$ and $\zeta_n = \eu^{2\iu\pi/n}$. The $n$-th cyclotomic polynomial is $\Phi_n = \prod_{\gcd(k,n)=1,\ 1 \le k \le n} \bigl(X - \zeta_n^k\bigr)$, of degree $\varphi(n)$; grouping the roots of $X^n - 1$ by exact order, $X^n - 1 = \prod_{d \mid n}\Phi_d$, which shows inductively that $\Phi_n \in \Z[X]$ (Euclidean division of monic integer polynomials).