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.

Useful Theorem in Introductory Ring Theory

Something interesting I realised in my studies in Math is that certain theorems are more “useful” than others. Certain theorems’ sole purpose seem to be an intermediate step to prove another theorem and are never used again. Other theorems seem to be so useful and their usage is everywhere.

One of the most “useful” theorems in basic Ring theory is the following:

Let R be a commutative ring with 1 and I an ideal of R. Then

(i) I is prime iff R/I is an integral domain.

(ii) I is maximal iff R/I is a field.

With this theorem, the following question is solved effortlessly:

Let R be a commutative ring with 1 and let I and J be ideals of R such that I\subseteq J.

(i) Show that J is a prime ideal of R iff J/I is a prime ideal of R/I.

(ii) Show that J is a maximal ideal of R iff J/I is a maximal ideal of R/I.

Sketch of Proof of (i):

J is a prime ideal of R iff R/J is an integral domain. (R/J\cong \frac{R/I}{J/I} by the Third Isomorphism Theorem. )\iff J/I is a prime ideal of R/I.

(ii) is proved similarly.