I'm trying to find a proof for the if-direction of the following theorem:
Let $F$ be a field of characteristic zero and $f(x)\in F[x]$. The Galois group of $f(x)$ is solvable if and only if $f(x)$ is a solvable polynomial (i.e. Its splitting field is contained in a radical extension of $F$).
Could you give me the main steps I need for a proof? I think I can figure out the details on my own then. Thank you!