Proving $\mathbb{Z}[i]^{\times}$ is finite I have no idea how to prove that $\mathbb{Z}[i]^{\times}$ is finite (unit group of $\mathbb{Z}[i]$).
I don't know any theorem and never studied properties of unit groups, so I don't even know where to start here.
This fact fact appeared to me as an example while I was studying modules.
Any help or hint would be appreciated. Thanks in advance.

Hint: $\frac{1}{a+bi}=\frac{a-bi}{a^2+b^2}$.  If $a,b\in \mathbb Z$, when is this still in $\mathbb Z[i]$?
