# Difference between revisions of "Cyclotomic polynomial"

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In [[algebra]], a '''cyclotomic polynomial''' is a [[polynomial]] whose roots are a set of primitive [[root of unity|roots of unity]]. The ''n''-th cyclotomic polynomial, denoted by Φ<sub>''n''</sub> has [[integer]] cofficients. | In [[algebra]], a '''cyclotomic polynomial''' is a [[polynomial]] whose roots are a set of primitive [[root of unity|roots of unity]]. The ''n''-th cyclotomic polynomial, denoted by Φ<sub>''n''</sub> has [[integer]] cofficients. | ||

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where μ is the [[Möbius function]]. | where μ is the [[Möbius function]]. | ||

==Examples== | |||

:<math>\Phi_1(X) = X-1 ;\,</math> | |||

:<math>\Phi_2(X) = X+1 ;\,</math> | |||

:<math>\Phi_3(X) = X^2+X+1 ;\,</math> | |||

:<math>\Phi_4(X) = X^2+1 ;\,</math> | |||

:<math>\Phi_5(X) = X^4+X^3+X^2+X+1 ;\,</math> | |||

:<math>\Phi_6(X) = X^2-X+1 ;\,</math> | |||

:<math>\Phi_7(X) = X^6+X^5+X^4+X^3+X^2+X+1 ;\,</math> | |||

:<math>\Phi_8(X) = X^4+1 ;\,</math> | |||

:<math>\Phi_9(X) = X^6+X^3+1 ;\,</math> | |||

:<math>\Phi_{10}(X) = X^4-X^3+X^2-X+1. ;\,</math> |

## Latest revision as of 14:55, 11 December 2008

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

The degree of is given by the Euler totient function .

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

By the Möbius inversion formula we have

where μ is the Möbius function.