I got the question from one of the previous old exam sheets, but I couldn't be sure to determine it. Let $\alpha=\sqrt[3]2$ and $\epsilon=e^{\frac{2\pi i}{3}}$(a primitive third root of 1) and let $K=\Bbb{Q}[\alpha,\epsilon]$. Find the intermediate fields of the extension $K$ over $\Bbb{Q}$.
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The Galois group is given by $S_3$, see here. By the Galois correspondence it is enough to find the subgroups of $S_3$. This has been solved here. Then you obtain all intermediate fields.
Edit: I found that the question was already answered here.
answered Apr 8, 2016 at 19:01
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$\begingroup$ ..It is not exactly the same, but I am trying to understand it by considering complex elements. $\endgroup$– UserAbApr 8, 2016 at 19:13
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$\begingroup$ It is exactly the same since $\epsilon=\zeta=e^{2\pi i/3}$. $\endgroup$ Apr 8, 2016 at 19:15
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