About a gaussian prime

Question

I have a question:

Let gcd$(a,b)=1$ and assume that $a+bi$ is a gaussian prime. How can we prove that $a^2+b^2$ is prime using the result found in here.

My idea: If I can show that $\Bbb Z[i]/\langle a+bi\rangle$ is a field then using this result, it follows that $\Bbb Z_{a^2+b^2}$ is a field and hence $a^2+b^2$ is prime. Am I correct or is there any simple way of proving it?

Answer 1

As you said yourself, if you can show that $\Bbb Z[i]/\langle a+bi\rangle$ is a field, you are done. But this is exactly the problem, so you are going in circles. A solution is given in this duplicate:

$a+bi$ is prime in $\mathbb{Z}[i]$ if and only if $a^2+b^2$ is prime in $\mathbb{Z}$

Also interesting for the converse is this question:

If $a^2 + b^2$ is a prime number $p$, with $p \equiv 1$ (mod $4$), then $a + bi$ is prime in the Gaussian integers $\mathbb{Z}[i]$