The Gaussian integers form a Euclidean domain

Statement

Let $\alpha$ and $\beta$ be Gaussian integers 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. 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 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.