## 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.

Add to solve laterLet $I$ be a nonzero ideal of the ring of Gaussian integers $\Z[i]$.

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

Add to solve laterDenote by $i$ the square root of $-1$.

Let

\[R=\Z[i]=\{a+ib \mid a, b \in \Z \}\]
be the ring of Gaussian integers.

We define the norm $N:\Z[i] \to \Z$ by sending $\alpha=a+ib$ to

\[N(\alpha)=\alpha \bar{\alpha}=a^2+b^2.\]

Here $\bar{\alpha}$ is the complex conjugate of $\alpha$.

Then show that an element $\alpha \in R$ is a unit if and only if the norm $N(\alpha)=\pm 1$.

Also, determine all the units of the ring $R=\Z[i]$ of Gaussian integers.