The Gaussian Integers are the set of complex numbers of the form , with 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 is any nonzero ideal in , then is finite.

We will need to use the fact that is an Euclidean domain, and thus also a Principal Ideal Domain (PID).

Thus for some nonzero . Let .

By the division algorithm, with or . We also note that .

Thus,

.

Since there are only finitely many elements with , thus is finite.