 Solution

Since $f(X)=X(\pi +X^{q-1})$, its iterates satisfyThe polynomial ${\Phi }_m$ is Eisenstein: it is monic, every nonleading coefficient is divisible by $\pi$, and its constant term is $\pi$. It is also separable, since $q-1$ is prime to the residue characteristic and the iterates have nonzero derivative.

If $\alpha \ne 0$ and $m$ is least with $f_m(\alpha )=0$, then ${\Phi }_m(\alpha )=0$. Eisenstein irreducibility makes ${\Phi }_m$ its minimal polynomial, so $K(\alpha )/K$ is totally ramified and separable. For $\alpha =0$ the extension is trivial and has the same properties. This is the Eisenstein layers of Lubin–Tate torsion argument.