= Solution
Since $f(X)=X(\pi+X^{q-1})$, its iterates satisfy
$$
\Phi_m(X):=\frac{f_m(X)}{f_{m-1}(X)}
=\pi+f_{m-1}(X)^{q-1}.
$$
The polynomial $\Phi_m$ is <Eisenstein polynomial>[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 $\alpha\ne0$ 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.
Back to article page
