Cube Roots of Unity

Cube Roots of Unity

## Definition / Concept

The cube roots of unity are the solutions of:

$$z^3=1$$

## Formula

The three cube roots are:

$$1,\omega,\omega^2$$

where:

$$\omega=e^{2\pi i/3}$$

$$\omega=-\frac12+i\frac{\sqrt3}{2}$$

$$\omega^2=-\frac12-i\frac{\sqrt3}{2}$$

## Important Identities

$$\omega^3=1$$

$$1+\omega+\omega^2=0$$

$$\omega\ne1$$

$$\omega^2+\omega+1=0$$

$$\frac1\omega=\omega^2$$

$$\frac1{\omega^2}=\omega$$

## Usage

- Used extensively in polynomial equations and complex-number simplification.