I have a Galois field extension $E/F$ of degree $p$ and $F$ has characteristic $p$ and contains all the roots of unity. I've trying to show that $E/F$ is not a solvable extension.
My main issue is that as $E/F$ is a prime degree extension then $Gal(E/F) = C_p$ and is therefore solvable? I must be missing something obvious or I have an incorrect definition of solvable.
Failing that, to show $E/F$ is not solvable I'd like to find the corresponding Galois field extensions of $F$ that are contained in $E$ (to find the normal subgroups of $Gal(E/F)$) but again as $p$ is a prime there are none?