What is $\mathbb F_2(\alpha,\beta)$?

Question

I would like to determine $\mathbb F_2(\alpha,\beta)$ where $\alpha$ and $\beta$ are elements of some extension $L \supset \mathbb F_2$ satisfying $\alpha^3+\alpha+1=0$ and $\beta^2+\beta+1=0$. So far, I have found that $\{1,\alpha,\alpha^2\}$ is a basis of $\mathbb F_2(\alpha)$ over $\mathbb F$ and that $\{1,\beta\}$ is a basis of $\mathbb F_2(\beta)$ over $\mathbb F$. Is there a way to combine those two results to fully determine $\mathbb F_2(\alpha,\beta)$?

Answer 1

The obvious candidate for a basis is $\{1, \alpha, \alpha^2, \beta, \alpha \beta, \alpha^2 \beta\}$. Are those linearly independent over $\mathbb F_2$? If not, $\beta \in \mathbb F_2(\alpha)$.