Fourth Roots of Unity

The fourth roots of unity solve $z^4 = 1$. Factoring: $z^4-1 = (z^2-1)(z^2+1) = (z-1)(z+1)(z-i)(z+i)$, giving four roots:

$z = 1, \; i, \; -1, \; -i$

These are exactly the four powers of $i$: $i^0, i^1, i^2, i^3$. Each one sits $90^\circ$ around the unit circle from the last.