I have to solve the following question:
Let $p(x) = x^6-7 \in \mathbb{Q}[x]$, then $Gal(p(x)) $ is a solvable group (the splitting field of $p(x)$ is contained in a radical extension). Give a series of normal subgroups of $Gal(p(x)) $ that satisfy the definition of a solvable group.
Is there a smart way to do this or do I have to calculate the Galois group and then search for correct normal subgroups?