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)<N(\alpha). 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)<N(\alpha)\}    \end{aligned}.

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

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