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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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