$$Gal(\overline{\Bbb{Q}}/(\overline{\Bbb{Q}}\cap \Bbb{Q}_p))\cong Gal(\overline{\Bbb{Q}}_p/\Bbb{Q}_p)$$ is pro-solvable (so that $\overline{\Bbb{Q}}/(\overline{\Bbb{Q}}\cap \Bbb{Q}_p)$ is a tower of radical extensions) so it is natural to ask:
Are there smaller fields $K\subsetneq \overline{\Bbb{Q}}\cap \Bbb{Q}_p$ whose absolute Galois group is still pro-solvable, ie. such that $\overline{\Bbb{Q}}/K$ is radical?
The next question is to ask for the other kind of fields of algebraic numbers minimal for the property "$\overline{\Bbb{Q}}/K$ is radical".
$Gal(\overline{\Bbb{Q}}_p/\Bbb{Q}_p)$ is pro-solvable because if $\Bbb{Q}_p(\zeta_{p^\infty-1},p^{1/(p^\infty-1)})\subset L\subset \overline{\Bbb{Q}}_p$ then any finite Galois extension $L'/L$ is totally ramified of degree $p^m$, we have $L'=L(a)$ where the $L[x]$-minimal polynomial of $a$ and the coefficients $\sigma(a)=\sum_{j=0}^{p^m-1}c_j a^j$ of its conjugates are in a finite extension $E/\Bbb{Q}_p$, taking the largest $d$ such that any automorphism $\sigma$ sends $\pi_{E(a)}$ to $\pi_{E(a)}+z(\sigma)\pi_{E(a)}^d+O(\pi_{E(a)}^{d+1})$ we get that $\sigma\to z(\sigma)$ is a non-trivial homomorphism $Gal(L'/L)\to O_{E(a)}/(\pi_{E(a)})$.