I need a check on the following question
Let $\alpha$ a primitive element of $\mathbb{F}_{2^n}$. Determine the degree of the minimal polynomial over $\mathbb{F}_2$. What can you say about the splitting field?
From theory I know that the degree $d$ of the min, poly is the minimum integer such that $\alpha^{2^d}=1$. Now, $\alpha$ is primitive, so this means that $\alpha^{2^n} = 1$, hence the the min. poly has degree $2^n$.
Then, since the min. poly is irreducible, by definition, I have that the splitting field of $h$ over $\mathbb{F}_2$ is given by $\mathbb{F}_{2^{2^n}}$
Is everything okay?