Cyclotomic polynomial

Definition. For $n\ge 1$, the cyclotomic polynomial $\Phi_n(x)\in\mathbb{Z}[x]$ is defined by

where the product runs over all primitive n-th roots of unity $\zeta$ in a fixed algebraic closure.

A fundamental factorization identity is

Over $\mathbb{Q}$, $\Phi_n(x)$ is irreducible , and its degree is $\varphi(n)$ (Euler totient).

See also. cyclotomic extension , splitting field .

Examples.

1. $\Phi_1(x)=x-1$, $\Phi_2(x)=x+1$.
2. $\Phi_3(x)=x^2+x+1$, since $x^3-1=(x-1)(x^2+x+1)$.
3. $\Phi_4(x)=x^2+1$, since $x^4-1=(x-1)(x+1)(x^2+1)$.