Galois criterion for solvability of a polynomial by radicals

Let $f\in F[x]$ be a polynomial over a field $F$, and let $K$ be its splitting field. If $K$ is a radical extension of $F$, then the Galois group $\mathrm{Gal}(K/F)$ is a solvable group.

Conversely, if the Galois group $\mathrm{Gal}(K/F)$ is a solvable group, then $K$ is a radical extension of $F$ provided that the characteristic of $K$ is either $0$ or greater than $\mathrm{deg}(f)$.