I am working with the ideals $\mathfrak{p}=\left<2,1+\sqrt{-5}\right>, \mathfrak{q}=\left<3,1+\sqrt{-5}\right>, \mathfrak{t}=\left<3,1-\sqrt{-5}\right>$ and I am trying to prove that they are prime in $\mathbb{Z}[\sqrt{-5}]$.
I understand a good method to do this involves taking norms of each element that generates the ideals. This gives, for instance, the norms of the generators of $\mathfrak{p}=4,6$. These are both divisible by $2$. I have also calculated $\mathfrak{p}^2=\left<2\right>$. I get the feeling that since both norms are divisible by the generator of the ideal squared this proves it is prime, but I am not quite sure why.
In a similar vein, the norms of the generators of $\mathfrak{q,t}=9,6$ which are both divisible by $3$. I know $\mathfrak{qt}=\left<3\right>$, but I am not sure if this is relevant, since it is not either ideal squared. I think the norm of the ideal itself being prime implies the ideal is prime, but I am not sure how to find the norm of the ideal from the norm of its generating elements.
This question is similar to: Prove that ideals are prime, but I don't quite understand the reasoning behind the chosen answer. I am not sure how the answerer deduces that the example there is prime either. Following their reasoning as far as I can, the fact that $2$ divides both norms means that $\left<2\right> \subset \mathfrak{p} \subset \mathbb{Z}[\sqrt{-5}]$ but I'm not sure how, and then I don't know how this proves it is prime.