Let $L/K$ be an extension of fields of characteristic zero. I want to prove that if $L^\times/K^\times$ is a torsion group (i.e. for every element $\alpha \in L$, some power of $\alpha$ lies in $K$), then $L=K$.

Reducing the the case where $L$ is of the form $L=K(\sqrt[p]{\alpha})$ seems appropriate. I have started sketching a proof that is sure to be very unpleasant, if successful. It's such a basic sounding fact that I was hoping someone else knows an easy proof.