Conditions for being an unramified extensions

Question

I'm not sure of the following: If I have a non-archimedean normed field $K$ of charateristic zero with residue field $k$ of characteristic $p>0$. If $\alpha$ is an element in some finite extension of $K$ such that there exists a non negative integer $h$ with $\alpha^h \in K$ then, $\alpha$ is an element of an unramified extension of $K$ ?

Answer 1

No. Let $K=\Bbb Q_2$, and $\alpha=\sqrt2$. Then $K(\alpha)/K$ is ramified.