Root of unity

In mathematics, a root of unity is a number some power of which is equal to one. An n-th root of unity is a number ζ such that ζn = 1. A primitive n-th root of unity is one which is an n-th root but not an m-th root for any m less than n. Any n-th root of unity is a primitive d-th root of unity for some d dividing n.

The n-th roots of unity are the roots of the polynomial Xn - 1; the primitive n-th roots of unity are the roots of the cyclotomic polynomial Φn(X).

Roots of unity are clearly algebraic numbers, and indeed algebraic integers. It is often convenient to identify the n-th roots of unity with the complex numbers exp(2π i r/n) with r=0,...,n-1 and the primitive n-th roots with those numbers of the form exp(2π i r/n) with r coprime to n. However, the concept of root of unity makes sense in other context such as p-adic fields and finite fields (in the latter case every non-zero element is a root of unity).