# Finding elements that are algebraic over a given field

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.

Hint: Let $$\theta=\sqrt[12]{2}$$. What is the degree of $$\theta^k$$, when $$k$$ divides $$12$$?
• Bonus points: What is the degree of $\theta^k$, when $k$ does not divide $12$? – lhf Aug 20 at 12:35