Let $F=F_0\subseteq F_1\subseteq ... \subseteq F_m=K$ be extensions with radicals, show that they exist extensions with radicals such that $F=K_0\subseteq K_1\subseteq ... \subseteq K_m=K$ where every $[K_i:K_{i-1}]$ is a prime number.
My first attempt was to use somehow solvable groups, but that doesn't work because if $[K_i:K_{i-1}]$ is cyclic doesn't mean it's of a prime order.
My second attempt was to show somehow that radical extensions can be manipulated so that the extension to always be $2$ and so prime.
Can someone give me a hint how to start, or the basic idea ?