# Primes lying above a prime in a Galois extension

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.

• The classification you are describing still holds even if $O_K[\theta] \subsetneq O_L$, provided that $p$ doesn't divide the conductor of $O_K[\theta]$ inside $O_L$ (this should be sufficient to consider primes that don't divide the discriminant of $f$, see here) – this misses only finitely many primes. See Neukirch's book on ANT, I.8.3, or also math.stanford.edu/~conrad/154Page/handouts/dedekindcrit.pdf. – Watson Dec 27 '18 at 20:51