Roots of unity sit evenly spaced around a circle
Some complex numbers return exactly 1 when raised to a whole-number power. The nth such roots mark n evenly spaced points on the unit circle, one of them always 1 itself. Named occasionally after Abraham de Moivre, they underpin number theory, group characters and the discrete Fourier transform.
Among the complex numbers, the nth roots of unity are given by the cosine plus i times the sine of 2k pi over n, as k runs from 0 to n minus 1. They include 1 always, and negative 1 whenever n is even. The same definition works in any field, or even any ring. In a field of characteristic zero the roots are algebraic integers; in fields of positive characteristic they live in a finite field, where every nonzero element is itself a root of unity.
A root is called primitive if no smaller positive power already gives 1. When n is prime, every nth root except 1 is primitive. In the trigonometric formula, the primitive ones are exactly those where k and n share no common factor, so there are as many of them as Euler's totient function counts. Sorting all n roots by the smallest power that returns them to 1 yields a classic identity: the totients of the divisors of n add up to n.
Arithmetic behaves tidily. Any whole-number power of an nth root is another nth root, and a reciprocal equals the complex conjugate. Exponents can be reduced modulo n, so only the remainder after division matters. Starting from a primitive root and taking successive powers visits every nth root exactly once before returning home.
Multiplying roots of unity or inverting them always produces more roots, so together they form an abelian group, the torsion part of the circle group. For fixed n the group is cyclic, generated by any primitive root, and the very term cyclic group comes from this link to the circle.
Source: Root of unity
