Let $D$ be a degree $3$ division algebra over a field $k$ of char not 2 and 3. Any such division algebra is cyclic. I am interested in knowing the cases when the reduced norm map $Nrd : D^* \rightarrow k^*$ is surjective. Of course, this happens over $\bar k$ and finite field etc. Here is my explicit question.
I want to relate surjectivity of reduced norm to the finiteness of $k^*/(k^*)^3$. To me it looks like not having enough of degree 3 field extensions is somehow responsible. I would appreciate examples, counterexamples or any reference in this direction.
Thanks a lot.