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I want to show that $p$ is unramified in a field extension $L/K$ if and only if it is unramified in the Galois closure $N/K$.

This is what I have so far: if $p$ is unramified in $N/K$, then because both ramification and residue degrees are multiplicative, and the only factors of 1 are 1, then $p$ is unramified in $L/K$. What I'm having trouble showing is when $p$ is unramified in $L/K$, it must also be unramified in $N/K$. My thinking is that since all the ramification/residue degrees are equal in the Galois closure, I just need to show one of each is 1. But how?

user404920
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    Are you familiar with the inertia group? Note that $N$ is the compositum of all the Galois conjugates of $L$. Moreover, all the Galois conjugates of $L$ are, by definition, isomorphic, so if $p$ is unramified in one of them, it is unramified in all of them. It remains to show that the compositum of two unramified fields is unramified. – Mathmo123 Feb 01 '21 at 22:42
  • Related: https://math.stackexchange.com/questions/1796472/ramification-of-prime-in-normal-closure (for splitting, see also https://math.stackexchange.com/questions/78091/splitting-of-primes-in-the-compositum-of-fields) – Watson May 18 '21 at 11:01

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