Let $K$ be a number field and $L/K$ be finite galois extension.
Let $p$ be a prime ideal of ring of integers of $K$.
Let $P$ be a prime ideal above $p$.
Let's think about corresponding local extension $L_P/K_p$, where $L_P$ and $K_p$ denotes completion of $L$ and $K$ with prime ideal $P$ and $p$.
My question is, $L/K$ is unramified at $p$ implies $L_P/K_p$ is
unramified ?
If there is only one prime above $p$, because completion of Dedekind domain keeps ramification degree and inertia degree, we can say corresponding local extension is umramified, but if there are more than 2 primes above $p$, I think we have no way to prove the statement.
But I'm having trouble finding counterexamples. This question asks concrete counterexample and the explanation why that forms counterexample,Thank you in advance.
Best Answer
In an extension $L/K$ of algebraic number fields (Galois or not), suppose a prime $p$ of $K$ factors as $pO_L=P_1^{e_1}\ldots P_g^{e_g}$. We say that $p$ is unramified in that extension if and only if all exponents $e_i$ are $1$.
From some little theorems in algebraic number theory, each extension $L_{P_i}/K_p$ of completions has ramification degree/index $e_i$.
So if the $L/K$ is ("globally") unramified over $p$, it is locally unramified at all primes lying over $p$.
Or do you mean to ask about something else here?