## Quotient Ring of the Gaussian Integers is Finite

The Gaussian Integers $\mathbb{Z}[i]$ are the set of complex numbers of the form $a+bi$, with $a,b$ integers. Originally discovered and studied by Gauss, the Gaussian Integers are useful in number theory, for instance they can be used to prove that a prime is expressible as a sum of two squares iff it is congruent to 1 modulo 4.

This blog post will prove that every (proper) quotient ring of the Gaussian Integers is finite. I.e. if $I$ is any nonzero ideal in $\mathbb{Z}[i]$, then $\mathbb{Z}[i]/I$ is finite.

We will need to use the fact that $\mathbb{Z}[i]$ is an Euclidean domain, and thus also a Principal Ideal Domain (PID).

Thus $I=(\alpha)$ for some nonzero $\alpha\in\mathbb{Z}[i]$. Let $\beta\in\mathbb{Z}[i]$.

By the division algorithm, $\beta=\alpha q+r$ with $r=0$ or $N(r). We also note that $\beta+I=r+I$.

Thus,

\begin{aligned}\mathbb{Z}[i]/I&=\{\beta+I\mid\beta\in\mathbb{Z}[i]\}\\ &=\{r+I\mid r\in\mathbb{Z}[i],N(r).

Since there are only finitely many elements $r\in\mathbb{Z}[i]$ with $N(r), thus $\mathbb{Z}[i]/I$ is finite.