Possiblities of dimensions of extension of two complex numbers of degree over rational numbers

Question

Let $\alpha$ and $\beta$ be complex numbers of degree 3 over $\mathbb{Q}$, and let $K = \mathbb{Q}(\alpha,\beta)$. Determine the possibilities for $[K:\mathbb{Q}]$.

We have $[K:\mathbb{Q}] = [\mathbb{Q}(\alpha):\mathbb{Q}][K:\mathbb{Q}(\alpha,\beta)]$, so $[K:\mathbb{Q}]$ can be $3,6,9$. For 3 example is just $\alpha = \beta = 2^{1/3}$, and for 9 square root of 3 and 2 will work. What about 6? Any ideas?

Answer 1

Take $\alpha=\sqrt[3]{2}$ and $\beta=j\sqrt[3]{2}$, where $j$ is a primitive third root of $1$.