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Find all $k \in \mathbb{N}$ such that there exist elements in the field $\mathbb{Q}(\sqrt[\leftroot{-2}\uproot{2}12]{2)}$ that are algebraic of order $k$ over $\mathbb{Q}$. For each such $k$ find an example of such an element.

Let $a$ be such an element. From the formula [$\mathbb{Q}:\mathbb{Q}(\sqrt[\leftroot{-2}\uproot{2}12]{2}]=[\mathbb{Q}:\mathbb{Q}(a)][\mathbb{Q}(a):\mathbb{Q}(\sqrt[\leftroot{-2}\uproot{2}12]{2})$ it follows that $k$ must be equal to 2, 3, 6, 12 or 4. I don't know where to go from here, nor do I know whether all such $k$ are viable. Any help would be appreciated.

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Hint: Let $\theta=\sqrt[12]{2}$. What is the degree of $\theta^k$, when $k$ divides $12$?

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  • $\begingroup$ Bonus points: What is the degree of $\theta^k$, when $k$ does not divide $12$? $\endgroup$ – lhf Aug 20 at 12:35

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