Suppose $L/K$ is a (finite) tamely ramified extension of local fields. What can be said about $\text{Gal}(L/K)$? For instance, is it true that $\text{Gal}(L/K)$ is the semidirect product of $I$ (the inertia subgroup, which we note is cyclic in this case) and $\text{Gal}(k_L/k_K)$?
I believe this is the case if $K=\mathbb{Q}_p$, where we can use Schur-Zassenhaus to the exact sequence $1\to I\to \text{Gal}(L/K)\to \text{Gal}(k_L/k_K)\to1$.
In general it is not a semi-direct product. Even over $\mathbb{Q}_p$.
Counter-example: This field of degree $9$ over $\mathbb{Q}_7$ has a cyclic Galois group, but it is obtained as an unramified cubic extension, followed by a tamely ramified cubic extension.
If the ramification index $e$ is coprime to the residue class degree $f$, then Schur-Zassenhaus implies that the Galois group is semi-direct.