Let be$$ \mathbb{Z} [ \sqrt{-2}]= \{ a+ b \sqrt{-2}] |a,b \in \mathbb{Z} \} $$
I want to find all units in $ \mathbb{Z} [ \sqrt{-2}] $
I found some similar questions..but I am still not sure how to work on that..
so take $ \phi(a+ b\sqrt{-2})= a^2 +2b^2 $
properties of the norm : $ |ab| = |a||b| $
so If there is an unit $ \alpha \in \mathbb{Z} [ \sqrt{-2}]$ than it must be $|\alpha \beta |=1= |\alpha| |\beta|$ so the only units would be $\alpha= -1 $ and $\alpha=1$?
I feel like this is not complete..would appreciate any help of you!!