Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$ Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$
Where $N$ is the norm function that maps $\gamma = a+b\sqrt{n} \mapsto \left | a^2-nb^2 \right |$
I have managed to prove $N(\gamma) = 1 \Rightarrow \gamma$ is a unit (i think), but cannot prove $\gamma$ is a unit $\Rightarrow N( \gamma ) = 1$
Any help would be appreciated, cheers

Hint $\rm\ \ unit\ \alpha\iff \alpha\:|\: 1\iff \alpha\alpha'\:|\:1 \iff unit\ \alpha\alpha',\ $ since $\rm\:\alpha\:|\:1\iff\alpha'\:|\:1' = 1$
