Cube Roots of Unity

The cube roots of unity are the solutions to $z^3 = 1$. Factoring $z^3 - 1 = (z-1)(z^2+z+1) = 0$ gives one real root and two complex ones:

$z = 1, \quad \omega = \frac{-1+i\sqrt{3}}{2}, \quad \omega^2 = \frac{-1-i\sqrt{3}}{2}$

$\omega$ is usually called the primitive cube root of unity — the other two roots, $1$ and $\omega^2$, can both be written as powers of it.