Let $K/k$ be an extension of number fields.
Is it true that for almost all places $v$ of $k$, the extension $K \otimes_k k_v / k_v$ is a cyclic extension of local fields? (Maybe under some additional assumption on $K/k$, e.g. if $K/k$ is Galois/abelian/...)
EDIT: OK, maybe that was not the question I wanted to ask. Is it true that for almost all paces $v$ of $k$, there exists a place $w$ of $K$ lying over $v$ such that the extension $K_w/k_v$ is cyclic?