TITLE: Totally ramified subextension in a finite extension of $\mathbf{Q}_p$
QUESTION [22 upvotes]: Let $K$ be a finite extension of $\mathbf{Q}_p$. Let $F_d$ be the unramified extension of $\mathbf{Q}_p$ of degree $d$. I would like to know whether there exists some $d \geq 1$ and some $L \subset K \cdot F_d$ such that $L/\mathbf{Q}_p$ is totally ramified and $K \cdot F_d / L$ is unramified. 
If $K/\mathbf{Q}_p$ is Galois, this can be done as follows. Take $d=[K:\mathbf{Q}_p]$. Write $Gal( F_d / \mathbf{Q}_p) = \langle \sigma \rangle$ and take any $\tilde{\sigma} \in Gal(K \cdot F_d / \mathbf{Q}_p)$ that lifts $\sigma$. Since $\tilde{\sigma}^d$ is the identity on both $K$ and $F_d$, it is the identity in $Gal(K \cdot F_d / \mathbf{Q}_p)$. We can then take $L = (K \cdot F_d)^{\langle \tilde{\sigma} \rangle}$.
Is the result still true if $K/\mathbf{Q}_p$ is not Galois?

REPLY [3 votes]: I'm a bit late but here are a few remarks on Kevin's example :

There is a unique $\mathfrak{A}_4$-extension of $\mathbf{Q}_2$ because there is a unique cyclic cubic extension $C$ (namely the unramified one), the group $G=\mathrm{Gal}(C|\mathbf{Q}_2)=\mathbf{Z}/3\mathbf{Z}$ has a unique irreducible degree-$2$ $\mathbf{F}_2$-representation $\rho$, and $\rho$ occurs with multiplicity $1$ in $C^\times/C^{\times2}$.  If $G$ acts on $D\subset C^\times/C^{\times2}$ through $\rho$, then Kevin's $M$ is $M=C(\sqrt D)$.
Kevin's $K$ is not galoisian over $\mathbf{Q}_2$ (its galoisian closure is $M$) and nor is any of its unramified extensions, so no  such $L$ exists (because an unramified extension of a quadratic extension is always galoisian). 
You can find $\mathfrak{A}_4$-extensions of ramification index $4$ and residual degree $3$ over every local field $F$ with finite residue field of characteristic $2$ (although they might not be unique, and in fact there are infinitely many of them if $F$ has characteristic $2$), so the argument can be made to work over every such $F$.
$\mathfrak{A}_4$-extensions are galoisian closures of primitive quartic extensions (as are $\mathfrak{S}_4$-extensions, and these are the only two possibilities).  I'm confident that the same trick can be applied over local fields with finite residue field of characteristic $p$ (arbitrary prime)  by working with primitive extensions of degree $p^2$ or perhaps $p^n$ for some $n$.  How does one find such extensions ?  See https://arxiv.org/abs/1608.04183.