# Primitive Complex Roots of Unity/Examples/Cube Roots

## Examples of Primitive Complex Roots of Unity

The primitive complex cube roots of unity are:

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

## Proof

There are $3$ (complex) cube roots of unity:

$1, \omega, \omega^2$

We have:

 $\ds \omega$ $=$ $\ds e^{2 \pi i / 3}$ $\ds \omega^2$ $=$ $\ds e^{4 \pi i / 3}$ $\ds \omega^3$ $=$ $\ds 1$ Cube Roots of Unity

Also:

 $\ds \omega^2$ $=$ $\ds e^{4 \pi i / 3}$ $\ds \paren {\omega^2}^2$ $=$ $\ds \omega^3 \times \omega$ $\ds$ $=$ $\ds 1 \times \omega$ Cube Roots of Unity $\ds$ $=$ $\ds \omega$ $\ds \paren {\omega^2}^3$ $=$ $\ds \paren {\omega^3}^2$ Cube Roots of Unity $\ds$ $=$ $\ds 1 \times 1$ Cube Roots of Unity $\ds$ $=$ $\ds 1$

Trivially, $1$ is not a primitive complex cube root of unity because you cannot make either $\omega$ or $\omega^2$ by multiplying $1$ by itself as many times as you like.

Hence the result.

$\blacksquare$