Difference between revisions of "Cyclotomic polynomial"

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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>

Revision as of 12:16, 9 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 Φn 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.

Examples