In some recent computation I came across certain cubic forms and was wondering about analogue of following result for quadratic forms.
If $k^*/(k^*)^2$ is finite then there are only finitely many quadratic forms up to equivalence. This can be easily seen because of diagonalisation.
I have been wondering if we can say following: given $k^*/(k^*)^3$ is finite is it true that there are only finitely many cubic forms up to equivalence. Or can I say something about being isotropic.