finite Galois group as a semidirect product of inertia and residue Galois group?

Question

Let $K/\mathbb{Q}_p$ be a finite extension, and $L/K$ be a finite Galois extension. We have a short exact sequence $$1 \rightarrow I_{L/K}\rightarrow \mathrm{Gal}(L/K) \rightarrow \mathrm{Gal}(k_L/k_K) \rightarrow 1$$

I was wondering which of these extensions have the property that $\mathrm{Gal}(L/K)$ is a semidirect product of $I_{L/K}$ and $\mathrm{Gal}(k_L/k_K)$. For instance, we know that if $e=|I_{L/K}|$ and $f=[k_L:k_K]$ are coprime then $G$ is a semidirect product, by Schur-Zassenhaus theorem.

Does anyone know a reference in which this question is discussed in details ?

Thanks,

Yoël.

Answer 1

Because of the Schur-Zassenhaus theorem that you mentioned, non split extensions $L/K$ should be looked for in the category of $p$-extensions (i.e. $G$ is a $p$-group). Using local CFT and the known structure of the Galois group of the maximal abelian pro-$p$-extension of $K$, it is not difficult to get hold of cyclic $p$-extensions $L/K$ which are ramified, but not totally ramified. If you want systematic families of such extensions, look e.g. at §6 of Luca Caputo's "A classification of the extensions of degree $p^2$ over $\mathbf Q_p$ ...", Journal de Théorie des Nombres de Bordeaux 19 (2007), 337–355, https://www.emis.de/journals/JTNB/2007-2/article02.pdf, where it is shown that there are exactly ($ p-1$) cyclic extensions of $\mathbf Q_p$ of degree $ p^2$ and ramification index $p$.

Answer 2

If we go to the algebraic closure, we get a short exact sequence (as above) $$1 \rightarrow I_K \rightarrow Gal(\overline{K}/K) \rightarrow \hat{\mathbb{Z}} \rightarrow 1,$$ and the preimage of $\mathbb{Z}$ is called the Weil group $\mathcal{W}(\overline{K}/K)$; i.e. $$1 \rightarrow I_K \rightarrow \mathcal{W}(\overline{K}/K) \rightarrow \mathbb{Z} \rightarrow 1$$ is exact.

Now the Weil group is dense in $Gal(\overline{K}/K)$ so for a finite Galois extension $L/K$ we have $\mathcal{W}(L/K) = Gal(L/K)$ so there is no issue in answering the question here.

By construction though, the Weil group is a semidirect product of the inertia group and a Frobenius element hence so is your Galois group.

I'm not sure of a reference for all this, but for Weil groups you could try looking at Tate's Number Theoretic Background.