# Difference between revisions of "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 Φn has integer cofficients.

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

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

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

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

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

By the Möbius inversion formula we have

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

where μ is the Möbius function.

## Examples

${\displaystyle \Phi _{1}(X)=X-1;\,}$
${\displaystyle \Phi _{2}(X)=X+1;\,}$
${\displaystyle \Phi _{3}(X)=X^{2}+X+1;\,}$
${\displaystyle \Phi _{4}(X)=X^{2}+1;\,}$
${\displaystyle \Phi _{5}(X)=X^{4}+X^{3}+X^{2}+X+1;\,}$
${\displaystyle \Phi _{6}(X)=X^{2}-X+1;\,}$
${\displaystyle \Phi _{7}(X)=X^{6}+X^{5}+X^{4}+X^{3}+X^{2}+X+1;\,}$
${\displaystyle \Phi _{8}(X)=X^{4}+1;\,}$
${\displaystyle \Phi _{9}(X)=X^{6}+X^{3}+1;\,}$
${\displaystyle \Phi _{10}(X)=X^{4}-X^{3}+X^{2}-X+1.;\,}$