Let $E$ be an field extension of $F$ and $\alpha \in E$. Determine $[F(\alpha) : F(\alpha^3)]$.
I'm unsure how to approach this problem because I thought I would try to test some examples and see what I get. I first tried with $\alpha = \sqrt{2}$ so $\mathbb{Q(\sqrt{2}})$ and $\mathbb{Q(2\sqrt{2}})$ in which case $$[\mathbb{Q(\sqrt{2}}) : \mathbb{Q(2\sqrt{2}})]=1$$ And so I thought the answer would be $1$ however, another example using $\alpha = 2^{1/3}$ we get $\mathbb{Q(2^{1/3})}$ and $\mathbb{Q(2)}$ in which case $$[\mathbb{Q(2^{1/3})}:\mathbb{Q(2)}]=3$$ And so now i'm unsure what the answer is, any help is appreciated thanks.
Hint $\alpha$ is a root of $X^3-\alpha^3$, therefore its minimal polynomial is a divisor of this.