Definition:Cyclotomic Polynomial

Definition
Let $n$ be a natural number greater than $0$.

The $n$th cyclotomic polynomial is the polynomial
 * $\displaystyle \Phi_n \left({x}\right) = \prod_\zeta \left({x - \zeta}\right)$

where the product runs over all primitive complex $n$th roots of unity, that is, those whose order is $n$.

From Condition for Complex Root of Unity to be Primitive it is seen that this can be expressed as:
 * $\displaystyle \Phi_n \left({x}\right) = \prod_{\substack {1 \mathop \le k \mathop \le n \\ \gcd \left\{ {k, n}\right\} = 1} } \left({x - \exp \left({\dfrac {2 \pi i k} n}\right)}\right)$