Let $\Bbb Z_p[i]:=\{a+bi\;:\; a,b \in \Bbb Z_p\,\,,\,\, i^2 = -1\}$
-(a)Show that if $p$ is not prime, then $\mathbb{Z}_p[i]$ is not an integral domain.
-(b)Assume $p$ is prime. Show that every nonzero element in $\mathbb{Z}_p[i]$ is a unit if and only if $x^2+y^2$ is not equal to $0$ ($\bmod p$) for any pair of elements $x$ and $y$ in $\mathbb{Z}_p$.
(a)I think that I can prove the first part of this assignment. Let $p$ be not prime. Then there exist $x,y$ such that $p=xy$, where $1
However, I don't know how to continue from here.