Cyclotomic polynomial - Imported Knowlpedia Content

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

\Phi_n(x) \;:=\; \prod_{\substack{1\le k\le n\\ \gcd(k,n)=1}} (x-\zeta_n^{\,k}),

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.

Remarks

A key structural identity is the factorization

x^n-1 \;=\; \prod_{d\mid n}\Phi_d(x),

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

[\mathbb{Q}(\zeta_n):\mathbb{Q}]=\deg \Phi_n=\varphi(n),

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

Examples

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

For an odd prime $p$ ,

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

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.