# Root of unity

In mathematics, a **root of unity** is an algebraic quantity some power of which is equal to one. An *n*-th root of unity is a number ζ such that ζ^{n} = 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 *X*^{n} - 1; the primitive *n*-th roots of unity are the roots of the cyclotomic polynomial Φ_{n}(*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(2πi.*r*/*n*) with *r*=0,...,*n*-1 and the primitive *n*-th roots with those numbers of the form exp(2πi.*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).