Norm on the Gaussian integers

Definition

The norm on the Gaussian integers is the map $N \colon \mathbb{Z}[i] \to \mathbb{N}$, $$N(a + b i) = a^2 + b^2 = (a + b i)(a - b i).$$

Remarks

The norm is multiplicative: $N(z w) = N(z) N(w)$ for all $z, w \in \mathbb{Z}[i]$. This follows from $|z w| = |z| |w|$ in $\mathbb{C}$. $N(z) = 0$ if and only if $z = 0$. The norm is a Euclidean function on $\mathbb{Z}[i]$. For example, $N(2 + i) = 5$ and $N(1 + i) = 2$.

Explanations

L1 human

The norm is the squared distance from $0$ to the point $a + b i$ in the complex plane. Because it is multiplicative, it turns questions about factoring Gaussian integers into questions about factoring integers.