# Definition:Cyclotomic Polynomial

## Contents

## Definition

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

The **$n$th cyclotomic polynomial** is the polynomial

- $\displaystyle \Phi_n \paren x = \prod_\zeta \paren {x - \zeta}$

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:

- $\displaystyle \Phi_n \paren x = \prod_{\substack {1 \mathop \le k \mathop \le n \\ \gcd \set {k, n} = 1} } \paren {x - \exp \paren {\dfrac {2 \pi i k} n} }$

## Examples

### First Cyclotomic Polynomial

The **first cyclotomic polynomial** is:

- $\map {\Phi_1} x = x - 1$

### Cyclotomic Polynomial of Prime Index

Let $p$ be a prime number.

The **$p$th cyclotomic polynomial** is:

- $\Phi_p \left({x}\right) = x^{p-1} + x^{p-2} + \cdots + x + 1$

## Also see

- Results about
**cyclotomic polynomials**can be found here.