How to prove that N(u) = 1 if and only if u is a unit in $Z[\sqrt-5]$ The norm of an element $u=a+b\sqrt-5$ in $Z[\sqrt-5]$ is defined as $N(u)= a^2 +5b^2$, now if $N(u) = 1$ then $a^2+5b^2 = 1$ but then how would i prove that it's a unit !?
Note that norm has the following property. $N(\alpha)=\alpha \bar{\alpha}$. Thus when $N(\alpha)=1$, this means $\alpha \bar{\alpha}=1$, hence $\alpha$ is a unit.
$N(u)=1$ implies $|a|=1$ and $b=0$, so $u=-1$ or $u=1$, hence $u$ must be a unit.