Cyclotomic polynomial

Fix an integer $n\ge 1$. In an algebraic closure of a field of characteristic $0$ (e.g. inside $\mathbb{C}$), an element $\zeta$ is a primitive n-th root of unity if $\zeta^n=1$ and $\zeta^d\ne 1$ for every proper divisor $d\mid n$. The $n$-th cyclotomic polynomial is

where $\zeta_n$ is any fixed primitive $n$-th root of unity. This definition is independent of the choice of $\zeta_n$, and $\Phi_n(x)\in\mathbb{Z}[x]$ is monic.

A key structural identity is the factorization

which can be taken as an equivalent recursive definition of $\Phi_n$ in $\mathbb{Z}[x]$. Over $\mathbb{Q}$, $\Phi_n(x)$ is irreducible, so it is the minimal polynomial of $\zeta_n$ and

linking cyclotomic polynomials to the cyclotomic extension $\mathbb{Q}(\zeta_n)$ and the splitting field of $x^n-1$.

Examples

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

2. For an odd prime $p$,

   $$\Phi_p(x)=1+x+x^2+\cdots+x^{p-1}.$$

3. Using $x^6-1=\prod_{d\mid 6}\Phi_d(x)$ and the known $\Phi_1,\Phi_2,\Phi_3$, one gets

   $$\Phi_6(x)=\frac{x^6-1}{(x-1)(x+1)(x^2+x+1)}=x^2-x+1.$$