7
$\begingroup$

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.

$\endgroup$
12
$\begingroup$

This was proved by I. Kaplansky, "A theorem on division rings", Canadian J. Math. 3 (1951), 290-292, see this link.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.