Cyclotomic polynomial

From Knowino

Jump to: navigation, search

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 $Φ n (X)$ 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.

[edit] Examples

$\Phi_1(X) = X-1 ;\,$

$\Phi_2(X) = X+1 ;\,$

$\Phi_3(X) = X^2+X+1 ;\,$

$\Phi_4(X) = X^2+1 ;\,$

$\Phi_5(X) = X^4+X^3+X^2+X+1 ;\,$

$\Phi_6(X) = X^2-X+1 ;\,$

$\Phi_7(X) = X^6+X^5+X^4+X^3+X^2+X+1 ;\,$

$\Phi_8(X) = X^4+1 ;\,$

$\Phi_9(X) = X^6+X^3+1 ;\,$

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

Some content on this page may previously have appeared on Citizendium.

Cyclotomic polynomial

[edit] Examples

Personal tools

Namespaces

Variants

Views

Actions

Search

Navigation

Community

Toolbox