Cyclotomic polynomial

In algebra, a cyclotomic polynomial is a polynomial whose roots are a set of primitive roots of unity. The n-th cyclotomic polynomial, denoted by Φ_n has integer cofficients.

For a positive integer n, let ζ be a primitive n-th root of unity: then

$\Phi _{n}(X)=\prod _{(i,n)=1}\left(X-\zeta ^{i}\right).\,$

The degree of $\Phi _{n}(X)$ is given by the Euler totient function $\phi (n)$.

Since any n-th root of unity is a primitive d-th root of unity for some factor d of n, we have

$X^{n}-1=\prod _{d|n}\Phi _{d}(X).\,$

By the Möbius inversion formula we have

$\Phi _{n}(X)=\prod _{d|n}(X^{d}-1)^{\mu (n/d)},\,$

where μ is the Möbius function.

Examples

$\Phi _{1}(X)=X-1;\,$

$\Phi _{2}(X)=X+1;\,$

$\Phi _{3}(X)=X^{2}+X+1;\,$

$\Phi _{4}(X)=X^{2}+1;\,$

$\Phi _{5}(X)=X^{4}+X^{3}+X^{2}+X+1;\,$

$\Phi _{6}(X)=X^{2}-X+1;\,$

$\Phi _{7}(X)=X^{6}+X^{5}+X^{4}+X^{3}+X^{2}+X+1;\,$

$\Phi _{8}(X)=X^{4}+1;\,$

$\Phi _{9}(X)=X^{6}+X^{3}+1;\,$

$\Phi _{10}(X)=X^{4}-X^{3}+X^{2}-X+1.;\,$