Suppose $K$ over $F$ is a field extension, and $\alpha \in K$.
My instructor says that "$\alpha$ is solvable over $F$ if there exists a radical extension $L$ of $F$ containing $\alpha$".
My question is : Is $L$ supposed to be a subfield of $K$? Or can $L$ be any field containing $F$ and $\alpha$?
In most books, $F$ is taken to be $\mathbb{Q}$ and $K$ is taken to be the field of algebraic numbers, so in that case such an $L$ (if exists) will be contained in $K$.
But I was wondering about the general case. Any feedback will be appreciated. Also, if possible, give a textbook reference for your answer.
Thank you in advance.