Unramified extension of local field is automatically galois because there is bijection between unramified extension of local field and extension of residue finite field, that is galois.
But, what about ramified extension of local field ?
Does every ramified extension of local field is galois ?
Best Answer
Well, at least as a placeholder, when $\mathbb Q_p$ contains no $n$th root of unity, then "the" extension $\mathbb Q_p(p^{1/n})$ is not Galois. It is of degree $n$, by Eisenstein's criterion, but is not "normal", because it is missing $n$th roots of unity, etc.
I would think that more complicated extensions would tend to have this property, as well, so this is not any sort of pathology.