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](http://mathoverflow.net/questions/61084/examples-of-galois-invariant-central-simple-algebras-which-arent-base-change).