# Definition:Root of Unity

## Definition

Let $n \in \Z_{> 0}$ be a strictly positive integer.

Let $F$ be a field.

The $n$th roots of unity of $F$ are defined as:

$U_n = \left\{{z \in F: z^n = 1}\right\}$

### Complex Roots of Unity

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

The complex $n$th roots of unity are the elements of the set:

$U_n = \set {z \in \C: z^n = 1}$

### Primitive Root of Unity

A primitive $n$th root of unity of $F$ is an element $\alpha \in U_n$ such that:

$U_n = \set {1, \alpha, \ldots, \alpha^{n - 1} }$

### Order of Root of Unity

Let $z \in U_n$.

The order of $z$ is the smallest $p \in \Z_{> 0}$ such that:

$z^p = 1$