the example of field extension over $\mathbb{Q}_p$ such that the Galois group is not solvable

Question

I would like to know the example of field extension over $\mathbb{Q}_p$ such that the Galois group is not solvable.

It is well known that totally ramified extension of local fields corresponds with inertia group, thus the Galois group of the extension is solvable group.

So we need to seek some extension which is not totally ramified.

Thank you in advance.

Answer 1

For a normal extension $E/F$ let $q=|O_E/(\pi_E)|$ then $F(\zeta_{q-1})/F$ is unramified (abelian) and $E/F(\zeta_{q-1})$ is normal and totally ramified.