Proving a specific simple field extension is closed under conjugation

Question

I want to show the the simple field extension of the rationals: $Q(2^{1/3}e^\frac{2\pi i}{3})$ is closed under conjugation. I know that all the needs to be shown is that $e^\frac{2\pi i}{3}\in Q(2^{1/3}e^\frac{2\pi i}{3})$, but I cannot seem to show it. I have given it some time as it should be relatively straightforward but it is just not coming.

Answer 1

You only have to prove that $\color{red}{2^{1/3}}\mathrm e^{\color{red}-\tfrac{2\pi i}{3}}\in Q\Bigl(2^{1/3}e^\tfrac{2\pi i}{3}\Bigr)$, which is by far easier.