---
tag: 06RH4XAGRY
kind: statement
role: proposition
name: Units of the Gaussian integers
msc: [11R11]
uses: [0K5W69AC0J, 03AN1V4ZFJ, 019NFGFPR5]
sources:
  - url: https://en.wikipedia.org/wiki/Gaussian_integer
    locator: Basic definitions
---

## Statement

The [[units|019NFGFPR5]] of the [[Gaussian integers|0K5W69AC0J]] $\mathbb{Z}[i]$ are $1$, $-1$, $i$ and $-i$.

## Remarks

The proof uses the [[norm|03AN1V4ZFJ]]: if $u v = 1$, then $N(u) N(v) = 1$, so $N(u) = 1$. The only solutions of $a^2 + b^2 = 1$ in integers are $(\pm 1, 0)$ and $(0, \pm 1)$.
Conversely, an element of norm $1$ is a unit, because $(a + b i)(a - b i) = 1$.
The group of units is cyclic of order $4$, generated by $i$.

## Explanations

### L1 human

A unit must have norm $1$, so it lies on the unit circle.
The only points with integer coordinates on the unit circle are $1$, $i$, $-1$ and $-i$.