We know the following theorem from Galois theory:
Let $F$ be a field of characteristic $0$ and $f(x) \in F[x]$. Then $f(x)$ is solvable by radicals if and only if the Galois group of $f(x)$ is solvable.
For fields of characteristic $p > 0$ one direction of this theorem is not true. We could take $F = \mathbb{F}_p(t)$ and $f(x) = x^p - x - t \in F[x]$. Then the Galois group of $f(x)$ is $\mathbb{F}_p$ which is solvable, but $f(x)$ is not solvable by radicals.
What about the other direction? If $f(x) \in F[x]$ is solvable by radicals, is the Galois group of $f(x)$ solvable even when $F$ has characteristic $p > 0$?