Cyclotomic polynomial
In mathematics, the n {\displaystyle n} -th cyclotomic polynomial, for any positive integer n {\displaystyle n} , is the unique irreducible polynomial with integer coefficients that is a divisor of x n − 1 {\displaystyle x^{n}-1} and is not a divisor of x k − 1 {\displaystyle x^{k}-1} for any k < n {\displaystyle k<n} . Its roots are all n {\displaystyle n} -th primitive roots of unity e 2 i π k n {\displaystyle e^{2i\pi {\frac {k}{n}}}} , where k {\displaystyle k} runs over the positive integers up to n {\displaystyle n} and coprime to n {\displaystyle n} (where i {\displaystyle i} is the imaginary unit).