If an extension is locally Galois, is it Galois?

Question

Consider an extension of number fields $L/K$. Basically, my question is know whether this extension has a given property, if we assume that the completions at all finite places of the extension have the same property. More precisely,

Assume that for almost all finite prime $p$ of $K$, there is a finite prime $P$ of $L$ above $p$, such that $L_P / K_p$ is Galois. Does it follow that $L/K$ is Galois?

One could make two variations:

1. Make the stronger hypothesis that for all prime $p$ of $K$ and any prime $P$ above $p$, the extension of local fields $L_P / K_p$ is Galois.

2. Replace "Galois" by "abelian", both in the hypothesis and conclusion.

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Some thoughts:

1. Recall that the converse holds: if $L/K$ is Galois, then so is any completion.

2. We may write $L = K(a)$ for some $a \in L$. Then I think that for any prime $p$ of $K$, there is $P \mid p$ such that $L_P = K_p(a)$ (consider the minimal polynomial $f$ of $a$ over $K$, then $a$ is a root of some irreducible factor of $f$ in $K_p[T]$, and we have $L \otimes_K K_p \cong K_p[T]/(f) \cong \prod\limits_{P \mid p} L_P$).

   But in general, $K_p(a) \cap \overline{K}$ strictly contains $K(a)$, so if we pick any $\sigma : L = K(a) \to \overline{K}$ and try to extend it to $K_p(a)$ in order to use the assumption, we are stuck.

3. We cannot consider the variation "replace Galois by solvable". Indeed, any finite extension of local fields is solvable, while there are $S_5$-extensions of number fields.

4. For the variation 2., if we further assume that $L/K$ is Galois (not necessarily abelian), the question is just to deduce or not that $\mathrm{Gal}(L/K)$ is abelian, knowing that all the decomposition subgroups $D(P \mid p)$ are abelian.

Answer 1

Every finite unramified extension of local fields is Galois, and even cyclic (the Galois group is isomorphic to the Galois group of the residue field extension, which is an extension of finite fields). So $L_P/K_p$ is cyclic whenever $P$ is unramified. All but finitely many primes are unramified, so even if we assume that $L_P/K_p$ is cyclic for all but finitely many $P$, we cannot conclude anything about the extension $L/K$.