Let $k\subset K$ a field extension. Assume that $K$ is infinite. Then the index of $k^∗$ in $K^∗$, $[K^∗:k^∗]$
(as groups) cannot be finite. How can I prove this?
If $k$ is finite then I know how to solve this since, if $\mid k\mid=q$ then , if we assume that $[K^∗:k^∗]= n$ is not finite, we have that $K$ can be partitioned into $n$ disjoint sets with $q$ elements. So $\mid K\mid =nq$
not finite, therefore contradiction.
I dont know how to prove it for k is infinite as well.
