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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.

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    $\begingroup$ 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. $\endgroup$ – Watson Dec 27 '18 at 20:51

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