Let $K'/K$ be a finite extension of local discrete valued fields, and let $E/K$ be any extension of local discrete valued fields. Assume that $E\otimes_KK'$ is a field, and that $E\otimes_KK'/E$ is unramified. Is it true that $K'/K$ is unramified?
I think the answer should be yes, but unfortunately I cannot prove it. One can reduce to the case when $K'/K$ is totally ramified, and try to derive a contradiction from there.