Gaussian integer

Remarks

The set $\mathbb{Z}[i]$ is closed under addition, subtraction and multiplication, so it is a subring of $\mathbb{C}$. It is an integral domain. The units of $\mathbb{Z}[i]$ are $1$, $-1$, $i$ and $-i$. In the complex plane, the Gaussian integers are the points of the square lattice. The norm $N(a + bi) = a^2 + b^2$ links them to sums of two squares.