# Complex Roots of Unity/Examples/Cube Roots

## Example of Complex Roots of Unity

The complex cube roots of unity are the elements of the set:

$U_3 = \set {z \in \C: z^3 = 1}$

They are:

 $\displaystyle$  $\, \displaystyle e^{0 i \pi / 3} \,$ $\, \displaystyle =\,$ $\displaystyle 1$ $\displaystyle \omega$ $=$ $\, \displaystyle e^{2 i \pi / 3} \,$ $\, \displaystyle =\,$ $\displaystyle -\frac 1 2 + \frac {i \sqrt 3} 2$ $\displaystyle \omega^2$ $=$ $\, \displaystyle e^{4 i \pi / 3} \,$ $\, \displaystyle =\,$ $\displaystyle -\frac 1 2 - \frac {i \sqrt 3} 2$

The notation $\omega$ for, specifically, the complex cube roots of unity is conventional.

### Conjugate Form

The Cube Roots of Unity can be expressed in the form:

$U_3 = \set {1, \omega, \overline \omega}$

where:

$\omega = -\dfrac 1 2 + \dfrac {i \sqrt 3} 2$
$\overline \omega$ denotes the complex conjugate of $\omega$.

## Proof

 $\displaystyle z^3 - 1$ $=$ $\displaystyle \paren {z - 1} \paren {z^2 + z + 1}$ Difference of Two Cubes/Corollary $\displaystyle \leadsto \ \$ $\displaystyle z$ $=$ $\displaystyle 1$ $\, \displaystyle \text { or } \,$ $\displaystyle z^2 + z + 1$ $=$ $\displaystyle 0$

Then:

 $\displaystyle z^2 + z + 1$ $=$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle z$ $=$ $\displaystyle \dfrac {-1 \pm \sqrt {1^2 - 4 \times 1 \times 1} } {2 \times 1}$ Quadratic Formula $\displaystyle$ $=$ $\displaystyle -\frac 1 2 \pm i \frac {\sqrt 3} 2$ simplifying

$\blacksquare$