# Definition:Root of Unity/Complex/Primitive

## Definition

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $U_n$ denote the complex $n$th roots of unity:

$U_n = \set {z \in \C: z^n = 1}$

A primitive (complex) $n$th root of unity is an element $\alpha \in U_n$ such that:

$U_n = \set {1, \alpha, \alpha^2, \ldots, \alpha^{n - 1} }$

Equivalently, an $n$th root of unity is primitive if and only if its order is $n$.

## Examples

### Primitive Complex Cube Roots of Unity

The primitive complex cube roots of unity are:

 $\ds \omega$ $=$ $\, \ds e^{2 \pi i / 3} \,$ $\, \ds = \,$ $\ds -\frac 1 2 + \frac {i \sqrt 3} 2$ $\ds \omega^2$ $=$ $\, \ds e^{4 \pi i / 3} \,$ $\, \ds = \,$ $\ds -\frac 1 2 - \frac {i \sqrt 3} 2$

## Also see

• Results about the complex $n$th roots of unity can be found here.