radical extension

A radical tower is a field extension $L/F$ which has a filtration

$$F=L_0\subset L_1\subset \mathrm{\cdots }\subset L_n=L$$

where for each $i$, , there exists an element ${\alpha }_i\in L_{i+1}$ and a natural number $n_i$ such that $L_{i+1}=L_i({\alpha }_i)$ and ${\alpha }_i^{n_i}\in L_i$.

A radical extension is a field extension $K/F$ for which there exists a radical tower $L/F$ with $L\supset K$. The notion of radical extension coincides with the informal concept of solving for the roots of a polynomial by radicals, in the sense that a polynomial over $K$ is solvable by radicals if and only if its splitting field is a radical extension of $F$.