TITLE: Units of Gaussian integers
QUESTION [16 upvotes]: How can we show that $\pm 1, \pm i$ are the only units in the ring of Gaussian integers, $\mathbb Z[i]$?
Thank you.

REPLY [19 votes]: Alternatively, $\mathbb{Z}[i] = \{ a + bi : a,b \in \mathbb{Z}\}$. So the norm of $\mathbb{Z}[i]$ is $a^{2} + b^{2}$. Now, $a+bi$ is a unit if and only if $a+bi$ divides $1$. Hence $N(a+bi)=\pm 1. $
So, $a^{2} + b^{2} = \pm 1$
Since $a,b \in \mathbb{Z}$, the only solutions are $a=\pm 1$, $b=0$ and $a=0$, $b=\pm1$. So the units are $1, -1, i, -i$