Let $L / F$ be a field extension.
Then $L$ is a radical extension of $F$ iff there exist $\alpha_1, \ldots, \alpha_m \in F$ and $n_1, \ldots, n_2 \in \Z_{>0}$ such that:
$(1): \quad L = K \left[{\alpha_1, \ldots, \alpha_m}\right]$
$(2): \quad \alpha_1^{n_1} \in F$
$(3): \quad \forall i \in \N_m: \alpha_i^{n_i} \in F \left[{\alpha_1, \ldots, \alpha_{i-1}}\right]$