what I want to prove: if $a+bi$ （both a and b are nonzero）is gaussian prime,then $N(a+bi)$ is prime integer.
I know the fact that if $a+bi$ is gaussian prime ,then its norm must have the form $p^2$ or $p$. Then I should prove the former is impossible,provided that $a$ and $b$ are both nonzero.
aussming $N(a+bi)=p^2$ ,we could claim that $p$ is a gaussian prime,if not,then $p$ could be factored into two irreducible element,in which case $(a+bi)(a-bi)=$product of more than two irreducible element, which is impossible because of the property of UFD.
So $p$ must be gaussian prime,in which case we could use the unique factorization to prove $(a+bi)$ is associate to $p$,contradicting the assumption.
is my reasoning right or not?