I'm trying to prove that if $K\subset M\subset L$ and $M/K$ and $L/K$ are both radical extensions then $L/M$ is a radical extension. My lecturer has used this in a couple of proofs without justifying it. It seems like it's something that would be true but I have no idea how to put a formal proof together!
Note: There isn't a typo here - I'm not trying to prove the trivial result that if $M/K$ and $L/M$ are radical then $L/K$ is radical.
If I understand correctly, this is almost trivial.
Say $L = L_0 \supseteq L_1 \supseteq \dotsc \supseteq L_k = K$ is a radition extension, with every $L_{i - 1} / L_i$ being simple radical, i.e. $L_{i - 1} = L_i[\sqrt[d_i]{\pi_i}]$ for some $\pi_i \in L_i$.
Let $M$ be a middle extension between $L$ and $K$, and for each $i$, let $M_i$ be the subfield of $L$ generated by $M$ and $L_i$.
It is then obvious that $M_0 = L$, $M_k = M$, and $M_{i - 1}$ is generated by $L_i$, $M$, and $\sqrt[d_i]{\pi_i}$, hence $M_{i - 1} = M_i[\sqrt[d_i]{\pi_i}]$.
While it is possible that $M_{i - 1} = M_i$ for some $i$, this doesn't affect the fact that $L/M$ is a radical extension.
Note that there is no need to assume that $M/K$ is radical.