I am a high school student interested in studying mathematics at university. I've been doing some independent study and have found the Gaussian Integers $\mathbb{Z}[i]$ particularly interesting. I was thinking about primes in $\mathbb{Z}[i]$, and came across the following results:
First, considering which primes $p$ in $\mathbb{Z}$ remain prime prime in $\mathbb{Z}[i]$, I think the following is true:
- If $p \equiv 3 \bmod{4}$, then it remains prime in $\mathbb{Z}[i]$.
- If $p \equiv 1 \bmod{4}$, then $p$ is not prime in $\mathbb{Z}[i]$. This is because $p$ can be written as a sum of two squares: $p = a^2 +b^2$, and so factorizes in $\mathbb{Z}[i]$ as: $p = (a+ib)(a-ib)$, where neither $a+ib$ nor $a-ib$ are units (the latter is true because $p \neq 1$).
However, I realized that this is not the end of the story, as there could be primes in $\mathbb{Z}[i]$ of the form $a +ib$, where neither a nor b equals zero. In this case, the only result I could find was that if $a^2 + b^2$ is prime (e.g. in the case of $1+i$) then $a+ib$ is prime in $Z[i]$. But this does not determine all the primes in $\mathbb{Z}[i]$.
My question is: Is there a known necessary and sufficient criterion for a Gaussian Integer $a+ib$ to be prime in $\mathbb{Z}[i]$? If not, are there any additional primality tests known in $\mathbb{Z}[i]$?
Any help and guidance would be greatly appreciated.
