Field extension, primitive root of unity solvable by radicals

Question

I'm stuck with this exercise.

I'd like to show, that for a field $K$ with $\mathrm{char}(K)=0$ and $\omega_n$ being a primitive n-th root of unity $K(\omega_n)/K$ is solvable.

I thought I could do induction by n, but I didn't get that far with that.