---
tag: 0SPYEFH7H4
kind: statement
role: theorem
name: The Gaussian integers form a Euclidean domain
msc: [11R11]
uses: [0B268GVX95, 00HPT88ZX0, 0CHQKZFK5S]
sources:
  - url: https://en.wikipedia.org/wiki/Gaussian_integer
    locator: Euclidean division
  - url: https://en.wikipedia.org/wiki/Gaussian_integer
    locator: Historical background
---

## Statement

Let $\alpha$ and $\beta$ be [[Gaussian integers|0B268GVX95]] with $\beta \neq 0$.
Then there are Gaussian integers $q$ and $r$ with $\alpha = q \beta + r$ and $N(r) < N(\beta)$, where $N$ is the [[norm|00HPT88ZX0]].
So $\mathbb{Z}[i]$, with the norm as Euclidean function, is a Euclidean domain.

## Remarks

To find $q$, round the real part and the imaginary part of the complex number $\alpha / \beta$ to nearest integers. This gives even $N(r) \le \frac{1}{2} N(\beta)$.
As a consequence, $\mathbb{Z}[i]$ is a principal ideal domain and a unique factorization domain.
The quotient and the remainder are not always unique.
This is the analogue of the [[division algorithm|0CHQKZFK5S]] for integers.

## History

Carl Friedrich Gauss introduced the Gaussian integers in 1832, in his second paper on biquadratic reciprocity. In it, he proved unique factorization.

## Explanations

### L1 human

You can divide one Gaussian integer by another and get a remainder that is smaller in norm than the divisor.
So the Euclidean algorithm works in $\mathbb{Z}[i]$, and factorization into primes is unique, as for the integers.