The Quotient Ring $\Z[i]/I$ is Finite for a Nonzero Ideal of the Ring of Gaussian Integers

Problem 534

Let $I$ be a nonzero ideal of the ring of Gaussian integers $\Z[i]$.

Prove that the quotient ring $\Z[i]/I$ is finite.

Proof.

Recall that the ring of Gaussian integers is a Euclidean Domain with respect to the norm
$N(a+bi)=a^2+b^2$ for $a+bi\in \Z[i]$.
In particular, $\Z[i]$ is a Principal Ideal Domain (PID).

Since $I$ is a nonzero ideal of the PID $\Z[i]$, there exists a nonzero element $\alpha\in \Z[i]$ such that $I=(\alpha)$.
Let $a+bi+I$ be an arbitrary element in the quotient $\Z[i]/I$.
The Division Algorithm yields that
$a+bi=q\alpha+r,$ for some $q, r\in \Z[i]$ and $N(r) < N(\alpha)$.

Since $a+bi-r=q\alpha \in I$, we have
$a+bi+I=r+I.$ It follows that every element of $\Z[i]/I$ is represented by an element $r$ whose norm is less than $N(\alpha)$.

There are only finitely many elements in $\Z[i]$ whose norm is less than $N(\alpha)$.

(There are only finitely many integers $a, b$ satisfying $a^2+b^2 < N(\alpha)$.)

Hence the quotient ring $\Z[i]/I$ is finite.

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