The characterization of field extension radical solvability in terms of the Galois group is fairly straightforward and well-known. I'm curious about a more specific problem. Namely, given a field $F$ of characteristic $0$, a finite Galois extension $K/F$, and a prime $p$, when is $K$ solvable over $F$ using just $p$th radicals?
More precisely, when does there exist a chain of extensions:
$$F\leq F(c_1)\leq F(c_1,c_2)\leq \dots\leq F(c_1,c_2,\dots,c_n)$$
such that $c_i^p\in F(c_1,c_2,\dots,c_{i-1})$ for all $i$ and $K\leq F(c_1,c_2,\dots,c_n)$.
I believe that so far I have managed to show that if $K/F$ is finite + Galois and can be solved using $p$th radicals, then $\textrm{Gal}(K:F)$ has a normal solvable Sylow $p$-subgroup, $P$, such that $\textrm{Gal}(K:F)/P$ is cyclic and has order dividing $[F(\omega):F]$ (where $\omega$ is a primitive $p$th root of unity).
But I am not entirely sure if this is correct, and I don't know what a necessary+sufficient criterion would look like, the problem seeming to arise from the fact that $p$th roots of unity behave differently from $p$th roots in general and cannot be detected using Galois groups alone.