Cube Roots of Unity

The equation $z^{3}=1$ has three solutions in $\mathbb{C}$ — the cube roots of unity — with elegant properties used throughout algebra.

Finding the three roots

Solve $z^{3}=1$, i.e. $z^{3}-1=0$. Factor as a difference:

So either $z=1$, or $z^{2}+z+1=0$, whose roots (by the quadratic formula) are

The three cube roots of unity are therefore

The two complex roots are conjugates of each other, and either one is denoted $\omega$ (omega); its square is the other.

The two key properties

Properties of $\omega$

$\omega^{3}=1\qquad\text{and}\qquad 1+\omega+\omega^{2}=0.$

The first says $\omega$ is a genuine cube root of $1$ (so any power of $\omega$ reduces mod $3$: $\omega^{4}=\omega,\ \omega^{5}=\omega^{2}$, etc.). The second is just the coefficient relation from $z^{2}+z+1=0$; it lets you replace $1+\omega$ by $-\omega^{2}$, or $\omega+\omega^{2}$ by $-1$, to simplify expressions. …