Cyclotomic polynomial

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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

\Phi_n(X) = \prod_{(i,n)=1} \left( X - \zeta^i \right) .\,

The degree of Φn(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. \,
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