Given a galois extension of number fields $L/K$ of even degree, set $n\_0=\text{lcm} (\{[L\_v:K\_v] : v \in M\_K \})$ ($L\_v$ is any completion corresponding to a place dividing $v$).


Does $2$ divide $n\_0$?


This comes up in [this question](https://mathoverflow.net/questions/61084/examples-of-galois-invariant-central-simple-algebras-which-arent-base-change).



Can an even degree galois extension complete p-adically to an even galois extension
If I'm not mistaken, the slightly stronger result is true, that the lcm of orders of Frobenius elements must be even (forget ramified primes, that is). Isn't that a corollary of the Chebotaryov density theorem (there is a Frobenius in every conjugacy class, and some such class contains elements of even order)?


