# Cyclic extension of local fields

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?

• The tensor product $K \otimes_k k_v$ need not be a field for most $v$. Try $k = {\mathbf Q}$ and $K$ a Galois extension of ${\mathbf Q}$ whose Galois group is not cyclic (e.g., $K = {\mathbf Q}(\sqrt{2},\sqrt{3})$, whose Galois group is abelian). For all unramified places $v$ of $\mathbf Q$, the tensor product $K \otimes_{\mathbf Q} {\mathbf Q}_v$ is not a field since $v$ has at least two places over it (an unramified place with just one place above it has a decomposition group that is both cyclic and equal to the Galois group, which would be impossible if the Galois group is not cyclic). – KCd Jan 9 '13 at 2:03