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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.

Bias of Priene
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  • Thanks. And, like I said, I REALLY don't know where to start, I think I never proved that a group is finite in my life. – Bias of Priene Jul 24 '18 at 06:26
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    Consider norm function $N: \mathbb Z[i] \rightarrow \mathbbZ$, $N(a+bi) = a^2 + b^2$... Have you heard of this before? – cat Jul 24 '18 at 06:28

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

Aaron
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    Here is an alternative hint. Define $N(\alpha)=\alpha \overline{\alpha}$. Then $N(\alpha)$ is real, and $N(\alpha\beta)=N(\alpha)N(\beta)$. What happens if $\alpha\beta=1$? – Aaron Jul 24 '18 at 07:09