Question Suppose we have a finite cyclic Galois extension of fields $L/K$. Assume that for every prime $v$ of $K$ and any prime $w$ of $L$ over $v$ we have $L_w\simeq K_v$. I want to show that then $L\simeq K$.
Ideas I am trying to show it by elementary means. I have tried to use the fact that the Galois group of the completions is isomorphic to the decomposition group. And this would mean that this group is trivial. If this group is trivial, then the number of primes above any prime in $K$ is $[L:K]$, therefore the ramification indices and inertial degrees are both $1$ for any prime.
I feel like I am on the right track, because it seems that if both this quantities are one, then the primes $w$ are the same to the $v$? I am not sure how to follow from this. Any help is appreciated. Comment: I have realised I have not used the fact that $L/K$ is cyclic in my approach, maybe this can help?