I am attempting to do this using the minimal polynomial somehow. We have the minimal polynomial of $\sqrt{3}$ in the rationals is degree $2$ but the minimal polynomial of $2^{1/3}$ is degree 3. Does this mean that if $\sqrt{3}$ is in the given field its minimal polynomial must divide the minimal polynomial of $2^{1/3}?$ And in our case it does not, so this is a contradiction?
I am not sure if this is the right thought process, but for some reason I thought the minimal polynomial of everything in our extension field had same degree, I might be mixing this up with splitting fields.