$N(s) = 1$ if and only if s is a unit if and only if $s = \pm 1$ or $\pm i$. $s \in \mathbb{Z}[i]$

Question

I'm confused. This is a homework question and I am not asking for the answer, but I was just wondering if I have to prove the second iff then use that to prove the first iff. Either that or my professor is confused.

Answer 1

If $ z=x+iy \in\mathbb{C} $
then $ N[s]=1 \Rightarrow x^{2}+y^{2}=1 $

$\Rightarrow (x^{2}=1$ and $y^{2}=0)$ or $ (x^{2}=0$ and $y^{2}=1)$

Because x,y are integer.

If $x^{2}=1 \Rightarrow x=1$ or $x=-1$

Thus, $z_{1}=1$ or $z_{2}=-1$

If $y^{2}=1 \Rightarrow y=1$ or $y=-1$

Thus, $z_{3}=i$ or $z_{4}=-i$