Express the ideal $(6) \subset \mathbb Z\left[\sqrt {-5}\right]$ as a product of prime ideals.
I know I can write $(6)=(2)(3)=\left(1+\sqrt {-5}\right)\left(1-\sqrt {-5}\right)$. But I guess these factors might not be prime.
What's more, how to solve this kind of problem more systematically?