A complex number of the form $a+bi$ , where $a,b\in\mathbb{Z}$ , is called a Gaussian integer.

It is easy to see that the set $S$ of all Gaussian integers is a subring of $\mathbb{C}$ ; specifically, $S$ is the smallest subring containing $\{1,i\}$ , whence $S=\G$ .

$\G$ is a Euclidean ring, hence a principal ring, hence a unique factorization domain.

There are four units (i.e. invertible elements) in the ring $\G$ , namely $\pm 1$ and $\pm i$ . Up to multiplication by units, the primes in $\G$ are

• ordinary prime numbers $\equiv 3\mod 4$
• elements of the form $a\pm bi$ where $a^2+b^2$ is an ordinary prime $\equiv 1\mod 4$ (see Thue's lemma)
• the element $1+i$ .

Using the ring of Gaussian integers, it is not hard to show, for example, that the Diophantine equation $x^2+1=y^3$ has no solutions $(x,y)\in\Z\times\Z$ except $(0,1)$ .