What is solvable by radicals?

An extension $L/K$ (characteristic $0$ throughout this section) is radical if there is a tower $K = F_0 \subseteq \dots \subseteq F_r = L$ with $F_{i+1} = F_i(\alpha_i)$, $\alpha_i^{n_i} \in F_i$: each step adjoins an $n_i$-th root. A polynomial $P \in K[X]$ is solvable by radicals if its splitting field is contained in some radical extension of $K$.