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.



$L^\times / K^\times$ torsion $\Rightarrow L = K$?
This was proved by I. Kaplansky, "A theorem on division rings", Canadian J. Math. 3 (1951), 290-292, see this [link](http://cms.math.ca/cjm/v3/cjm1951v03.0290-0292.pdf).


