Note: here there is a good amount of rehash of this

recent post of mine on a related topic.

Okay so before I was trying to research what primes in a quadratic field $\displaystyle \mathbb{Q}(\sqrt{D})$ were like with respect to that field (since every field is an integral domain, and prime and irreducible elements are defined in integral domains), and unfortunately I missed a very basic piece of number theory and not surprisingly could not dig up much in my research precisely because there are no such primes; every non-zero element in a field is a unit, therefore every non-zero element divides every other element. So when we talk of primes in a quadratic field $\displaystyle \mathbb{Q}(\sqrt{D})$, what we really mean is the primes in its ring of integers, in particular for this thread, the Gaussian integers $\displaystyle \mathbb{Z}[i]$, the Eisenstein integers $\displaystyle \mathbb{Z}[\omega]$ where $\displaystyle \omega\equiv\frac{1}{2}(-1+i\sqrt{3})$, and $\displaystyle \mathcal{O}_{\mathbb{Q}(\sqrt{-5})}$. (It seems we can alternately define $\displaystyle \mathcal{O}_{\mathbb{Q}(\sqrt{-3})}$ as $\displaystyle \mathbb{Z}[\omega]$ with $\displaystyle \omega = \frac{1}{2}(1+i\sqrt{3})$;

reference.)

In any case there are several

Gaussian primes with small norms listed at MathWorld, likewise

here for Eisenstein primes. For more investigation I believe

this article on arXiv is directly applicable. I think rather than re-inventing the wheel in this thread and

this other one, it would be wise to take the time to read that paper and implement the algorithm, although it could also be fun to try to invent a new one (for primes and/or irreducibles), even if it is unlikely to be as efficient or clever, however my time is limited. I hope you feel satisfied with this response, but do post any further questions.