Definition:Cyclotomic Polynomial

Definition
Let $n \ge 1$ be a natural number.

The $n$th cyclotomic polynomial is the polynomial
 * $\ds \map {\Phi_n} x = \prod_\zeta \paren {x - \zeta}$

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:
 * $\ds \map {\Phi_n} x = \prod_{\substack {1 \mathop \le k \mathop \le n \\ \gcd \set {k, n} = 1} } \paren {x - \map \exp {\frac {2 \pi i k} n} }$

Also see

 * Cyclotomic Polynomial has Integer Coefficients