My script states that there are $2$ non-trivial subfields of $\mathbb{Q}(\zeta_7)$, where $\zeta_7$ is a $7$-th primitive root of unity. These subfields are $\mathbb{Q}(\zeta_7 +\overline \zeta_7)$ and $\mathbb{Q}(\sqrt{-7})$.
Definition of radical extension I am using:
An extension $L/K$ is a radical extension, if $\exists\gamma_1,...,\gamma_r\in L, n_1,...,n_r\in\mathbb{N}$, such that $L=K(\gamma_1,...,\gamma_r)$ and $\gamma_i^{n_i}\in K(\gamma_1,...,\gamma_{i-1})\ \forall i$
Ok, so using the definition on $\mathbb{Q}(\zeta_7)$ I obviously have the first condition satisfied and the second with $\zeta_7^7=1\in\mathbb{Q}$.
I am not sure what I don't understand at this point. Also I am unsure of why the subfields are even mentioned. In my script the lattice of the subfields is drawn at it says (translated from german): "Not a radical extension because $\zeta_7$ is only present at the the top of the lattice". I have no idea what this means.