Assuming $L$ is Galois over $K$, and that $O_L$ and $O_K$ are their respective rings of integers. Let $p$ be a prime ideal of $O_K$, is there a classification of the prime ideals laying above $p$ in $O_L$?
I understand that if $O_L$ is generated by a unique element over $O_K$, i.e. $O_L=O_K[\theta]$, then there exists such a description, using the irreducible prime factors of $\theta$'s integral dependence equation, after "dividing by $p$". However, as there is no particular reason for this to happen in general, I am curious as to whether there exists a more general classification.