when doing an exercise concerning field extensions, i've found one that stumped me. The question is that, given $\beta \in \mathbb{C}$, it is the case that the field extension $\mathbb{Q}(\beta^3) \subset \mathbb{Q}(\beta)$ is at most of degree 3, with all lower degrees occuring for some $\beta$. Now I have found examples for degree 2 ($e^\frac{2\pi\cdot i}{3}$ does the trick, its minimal polynomial is the second cyclotomic) and ofcourse 1, but I am unable to find one for degree 3. Also the inequality itsself is puzzling to me, I have a strong suspicion that one could try showing this using $\beta$'s minimal polynomial over $\mathbb{Q}$, but I have no idea where to start. Any help would be greatly appreciated!
2026-04-25 12:19:59.1777119599
$[\mathbb{Q}(\beta) : \mathbb{Q}(\beta^3)] \leq 3$
72 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in ABSTRACT-ALGEBRA
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