root of unity

A root of unity is a complex number that, when raised to some whole-number power, gives exactly 1. The name says it plainly: a root (an n-th root) of unity (the number one). Over the real numbers, 1 has only one or two such roots, but over the complex numbers the picture is far richer and more symmetric.

Formally, an n-th root of unity is a solution of z^n = 1. There are exactly n of them, and they are spaced perfectly evenly around the unit circle, forming the vertices of a regular n-sided polygon. Using polar form, they are the numbers cos(2πk/n) + i sin(2πk/n) for k = 0, 1, ..., n-1. One of them is always 1 itself, sitting at angle 0.

Roots of unity are everywhere in mathematics — they power the discrete Fourier transform, signal processing, and the deep symmetry arguments of Galois theory. A primitive n-th root of unity is one whose powers generate all n of them, which makes the set a cyclic group under multiplication.