My question is:
How can I find the prime (ideal) divisors of 2 and 3 in the ring of integers of $\mathbb Q[\sqrt{-14}]$ and $\mathbb Q[\sqrt{-23}]$?
Here's what I have so far.
I found that (2, $\sqrt{-14}$) and (2, -$\sqrt{-14}$) are bases for the prime divisors of (2) in the ring. It also seems like (3, 1 +/- $\sqrt{-14}$) are the divisors of (3) in the ring. I can show that the divisors of (2) are prime, since the product of any two elements in the ring being in the ideal implies that one is in the ideal. However, I'm not sure how to show that the divisors of 3 are prime, and I'm not sure how to deal with the Q[$\sqrt{-23}$] case, where -23 is congruent to 1 (mod 4).
Thank you.