I have here defined:
$L/K$ is a radical extension iff there is a tower of fields $K=K_0\subseteq K_1\subseteq \cdots \subseteq K_m = L$ with $K_{i+1}=K_i(\alpha_i)$ and $\alpha_i^{n_i} \in K_i$ where $n_i\in \Bbb N$
Firstly: Can I have some examples?
Secondly: This means that for everything we adjoin, we have some power of it already in the field, so we may have $\Bbb Q(\sqrt[4]{2})/\Bbb Q$ is a radical extension, since $\sqrt[4]{2}^4=2\in \Bbb Q$ and also, if we had:
$\Bbb Q\subseteq \Bbb Q(\sqrt{2})\subseteq \Bbb Q(\sqrt[4]{2})$, this is a radical extension, since $\sqrt{2}^2\in \Bbb Q, \sqrt[4]{2}^2\in \Bbb Q(\sqrt{2})$
Is this a correct interpretation?