---
tag: 0S61N75H45
kind: definition
name: Root of unity
msc: [11R18]
defines: 0YV2CKP2AV
uses: [09FTAET5Z4]
sources:
  - url: https://en.wikipedia.org/wiki/Root_of_unity
    locator: General definition
---

## Definition

Let $n$ be a positive integer.
A complex number $z$ is an $n$th root of unity when $z^n = 1$.
A complex number $z$ is a root of unity when it is an $n$th root of unity for some positive integer $n$.

## Remarks

The same definition works in any field, or in any commutative ring.
The $n$th roots of unity in $\mathbb{C}$ form a group under multiplication, written $\mu_n$.
For example, the $4$th roots of unity are $1$, $i$, $-1$ and $-i$.
Each root of unity $z$ satisfies $|z| = 1$, because $|z|^n = |z^n| = 1$.

## Explanations

### L1 human

A root of unity is a number that gives $1$ when you multiply it by itself often enough.
The number $-1$ is one, because $(-1)^2 = 1$, and so is $i$, because $i^4 = 1$.