Cyclotomic polynomial

In mathematics, more specifically in algebra, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients, which is a divisor of ${\displaystyle x^{n}-1}$ and is not a divisor of ${\displaystyle x^{k}-1}$ for any k < n. Its roots are all nth primitive roots of unity ${\displaystyle e^{2i\pi {\frac {k}{n}}}}$, where k runs over the positive integers not greater than n and coprime to n. In other words, the nth cyclotomic polynomial is equal to

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