Let $\pi$ be a prime element in $\Bbb{Z}[i]$. Then $N(\pi)=2$ or $N(\pi)=p$ s.t. $p$ is prime and $p\equiv 1\pmod 4$ 
Let $\pi$ denote a prime element in $\Bbb{Z}[i]$ such that $\pi\not\in \Bbb{Z},i\Bbb{Z}$. Prove that $N(\pi)=2$ or $N(\pi)=p$, where $p$ is a prime number $\equiv 1\pmod4$. Give a complete classification of the prime elements of $\Bbb{Z}[i]$ using the prime numbers in $\Bbb{Z}$.

This exact question has been asked here but I do not understand the answers given, so I will ask it again in hope of further explanation.
Here's everything I know that will probably help me with the problem:


*

*We know that $\pi=a+bi$ where $a,b\in \Bbb{Z}\setminus \{ 0\}$.

*Fermat's Two Square theorem: A prime number $p\equiv 1 \mod4$
satisfies $p=a^2+b^2$ where $a\neq b$ and $a,b\in \Bbb{N}\setminus
   \{0\}$.

*$\pi$ is irreducible in $\Bbb{Z}[i]$

*If $\pi=xy$ then $\pi \mid x$ or $\pi \mid y$

*If $N(\pi)=2$ then $a=\pm 1, b=\pm 1$.

*If $N(\pi)=p$ where prime $p\equiv 1\pmod 4$ then we know $p=a+bi$ for some $a,b\in \Bbb{Z}\setminus \{0\}$ by Fermat's theorem.


I can't figure out where to start. I was thinking of starting with this:
Let $x,y\in \Bbb{Z}[i]$ such that $x=x_1+x_2 i$ and $y=y_1+y_2i$. Then $\pi\mid x$ or $\pi\mid y$. Suppose $\pi\mid x$..... 
Can I have a hint on how to begin this?

The gaussian primes fall into three following classes
1.all positive rational primes of the form $4k+3$ and their associates.
2.the number $1+i$ and it's associates.
3.all associates to the numbers of the form $x+iy$ such that $x^2+y^2$ is a rational prime of the form $4k+1$.

The theory of algebraic numbers-Harry Pollard-p.17
