I'm struggling with the following past exam question, parts (b) and (c):
So, starting with (b): since $E/F$ is Galois of degree $p$, we know that $$\Gamma(E/F)\cong C_p\cong \langle\sigma\rangle,$$ where $\sigma\colon\lambda\mapsto\lambda^p$. Further, $$E=\mathrm{split}_F(x^{p^p}-x).$$ We know then that, if $\alpha\in E\setminus F$ then $\sigma^k(\alpha)\in E\setminus F$ for all $k=1,\ldots,p$, but I'm not too sure where to proceed from here. It seems like it might be an idea to consider the cases $q<p^p$, $q>p^p$, and $q=p$, and try to find some contradiction there by looking at an $F$-basis of $E$ containing powers of $\alpha$, but I can't seem to get anywhere far with that.
As for the second part, I'm really not too sure where to begin!
For (c): I know the definition of solvability for an extension (solvable Galois group, i.e. its Galois group contains a chain of normal subgroups where each quotient is abelian), but can't get anywhere on the last part of the question. What's really confusing me is that $E/F$ is Galois of degree $p$, and therefore must have Galois group $C_p$, which is solvable! So there must be some obvious mistake I'm making...
Full solutions would be great, but preferably in spoiler tags, with helpful hints in the answer, so I can at least try to solve it after some help!