Finding dimension of a field extension

Question

How would anyone go about this problem?

Find dim$_\mathbb{Q}\mathbb{Q}(\alpha,\beta)$ where $\alpha^{3}=2$ and $\beta^{2}=2$.

Thanks for your help, I really don't know how to go about this problem.

Answer 1

Hint: by the dimension formula, $$ [\mathbb{Q}(\alpha,\beta):\mathbb{Q}]= [\mathbb{Q}(\alpha,\beta):\mathbb{Q}(\beta)] [\mathbb{Q}(\beta):\mathbb{Q}] $$ (where $[F:K]=\dim_K F$). Is there a root of $X^3-2$ in $\mathbb{Q}(\beta)$?

Answer 2

Let $L=K(\alpha,\beta)$, you know that $K(\alpha)$ and $K(\beta)$ are both intermediate fields of L/K. Looking at $X^2 -2$ and $X^3 - 2$ and using the hint above, you know that [L:K] has to be at least 6, since 2 and 3 have to divide the degree. Why is the degree at most 6?