Cyclotomic Polynomial

Cyclotomic Polynomial

In algebra, the nth cyclotomic polynomial, for any positive integer n, is the unique polynomial with integer coefficients, which is a divisor of and is not a divisor of for any k < n. Its roots are the nth primitive roots of unity $e^{2i\pi\frac{k}{n}}$ , where k runs over the integers lower than n and coprime to n. In other words, the nth cyclotomic polynomial is equal to

$\Phi_n(x) = \prod_\stackrel{1\le k\le n-1}{\gcd(k,n)=1} (x-e^{2i\pi\frac{k}{n}})$

It may also be defined as the monic polynomial with integer coefficients, which is the minimal polynomial over the field of the rational numbers of any primitive root of unity ( is such a primitive root).

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