Definition 1 Let $K$ be a field of characteristic $0$. Let $p$ be a prime number. Let $\alpha$ be an element of an algebraic closure of $K$ such that $\alpha^p \in K$ and $X^p - \alpha^p$ is irreducible over $K$. Then we call $K(\alpha)/K$ a simple prime radical extension.
Definition 2 Let $K = K_0 \subset K_1 \subset \cdots \subset K_n = L$ be a tower of fields. If $K_i/K_{i-1}$ is a simple prime radical extension for $i =1, \dots, n$, we call $L/K$ is a prime radical extension.
Definition 3 Let $L/K$ be a field extension. If there exists a prime radical extension $E/K$ such that $L$ is a subfield of $E$, we call $L/K$ a prime radically solvable extension.
Lemma 1 Let $K \subset M \subset L$ be a tower of fields. Suppose $M/K$ and $L/M$ are prime radically solvable extensions. Then $L/K$ is also a prime radically solvable extension.
Proof: This is proved in Lemma 10 of my answer to this question.
Lemma 2 Let $L/K$ be a finite Galois extension of prime degree $p$. Suppose $char(K) = 0$ and $K$ has a $p$-th primitive root of unity. Then $L/K$ is a simple prime radical extension.
Proof: By this question, there is an element $\alpha$ of $L$ such that $L = K(\alpha)$ and $\alpha^p$ is an element of $K$. Since $p = [L\colon K]$, $X^p - \alpha^p$ is irreducible over $K$. Hence $L/K$ is a simple prime radical extension. QED
Lemma 3 Let $L/K$ be a finite Galois extension of prime degree $p$. Suppose $char(K) = 0$. Then $L/K$ is a prime radically solvable extension.
Proof: Let $\Omega$ be an algebraic closure of $L$. Let $\zeta$ be a primitive $p$-th root of unity in $\Omega$. Then $L(\zeta)/K(\zeta)$ is a Galois extension and $Gal(L(\zeta)/K(\zeta))$ is isomorphic to a subgroup of $Gal(L/K)$. Hence it is a cyclic group of order $p$ or $1$. By Lemma 2, $L(\zeta)/K(\zeta)$ is a prime radically solvable extension. On the other hand, by Lemma 6 of my answer to this question, $K(\zeta)/K$ is a prime radically solvable extension. Hence, by Lemma 1, $L(\zeta)/K$ is a prime radically solvable extension. Hence $L/K$ is a prime radically solvable extension. QED
Theorem Let $K$ be a field of characteristic $0$. Let $L/K$ be a finite Galois extension. Suppose $G = Gal(L/K)$ is sovable. Then $L/K$ is a prime radically solvable extension.
Proof: There exists a tower of subgroups of $G$: $G = G_n \supset G_{n-1} \supset \cdots \supset G_1 \supset G_0 = 1$, where each $G_i/G_{i-1}$ is a cyclic group of prime degree. Hence there exists a tower of fields: $K_0 = K \subset K_1 \subset \cdots \subset K_n = L$, where each $K_i/K_{i-1}$ is a cyclic Galois extension of prime degree. By Lemma 3, $K_i/K_{i-1}$ is a prime radically solvable extension. Hence, by Lemma 1, $L/K$ is a prime radically solvable extension. QED