---
tag: 0WX11EA3F3
kind: object
name: Group of nth roots of unity
aliases: [roots-of-unity]
msc: [20K01]
instance_of: [082GHNPJ53]
uses: [071DDFVG4J]
sources:
  - url: https://en.wikipedia.org/wiki/Root_of_unity
    locator: Group of nth roots of unity
---

## Description

Let $n$ be a positive integer.
The group of $n$th roots of unity is the set $\mu_n = \{z \in \mathbb{C} : z^n = 1\}$ with complex multiplication.
It is a finite [[abelian group|082GHNPJ53]] of order $n$, and its elements are $e^{2\pi i k / n}$ for $k = 0, 1, \ldots, n - 1$.
It is cyclic, and the element $e^{2\pi i / n}$ generates it.

## Remarks

The group $\mu_n$ is a subgroup of the [[circle group|04EAFVFTF7]].
For $n = 4$, the group is $\{1, i, -1, -i\}$.
A generator of $\mu_n$ is called a primitive $n$th root of unity, and there are $\varphi(n)$ of them, where $\varphi$ is Euler's totient function.

## Explanations

### L1 human

The $n$th roots of unity are $n$ points on the unit circle that cut it into $n$ equal arcs.
Multiplying two of them adds their angles, so the product is again one of these points.
They behave like the rotations of a regular polygon with $n$ sides.