In Marcus book “Number Fields” I have this exercise: (page 124, number 25) Let $L$ be a normal extension of $K$ and suppose $K$ contains a prime which becomes a power of a prime in $L$. Prove that the Galois group is solvable.
In the preceding exercise I have proved that $D$ and $E$ are solvable groups. If I’m not mistaken quotient of solvable groups is solvable. The the Galois group is isomorphic to the quotient $D/E,$ hence is solvable. I think there is something wrong in my reasoning, since if I’m right I don’t understand the sense of this exercise...