If $\omega$ is a non-real cube root of unity, find the value of $1 + \omega + \omega^2$.

The three cube roots of unity are $1, \omega, \omega^2$, and they are roots of $z^3 - 1 = (z-1)(z^2 + z + 1) = 0$. Vieta's formulas on $z^2 + z + 1$ give $\omega + \omega^2 = -1$, so $1 + \omega + \omega^2 = 0$. Equivalently, the sum of *all* $n$-th roots of unity is zero for $n \ge 2$.