Let $\alpha$ be a root of $g(X) = X^3 + X^2 + 1 \in \mathbb{F}_2[X]$ and $K = \mathbb{F}_2(\alpha)$. Let $f$ be an irreducible polynomial in $K[X]$ of degree 4. Let $\beta$ be a root of $f$ and let $L$ be a splitting field of $f$ over $K$.
Question 1: how many intermediate fields does $L/K$ have?
Question 2: why are all zeroes of $f$ of the form $\beta^k$ for some integer $k$?
My proffessor says those questions are closely related, so I ask them both in the same post.
I've already made some progress: as $g(0) = g(1) = 1 \neq 0$, $g$ is irreducible, so $K \cong \mathbb{F}_2/(g)$. Because every $x \in K$ can be written as $a + b\alpha + c\alpha^2$ with $a, b, c \in \mathbb{F}_2$, we have that $K$ has characteristic 2. The amount of intermediates fields is equal to the the amount of proper subgroups of $Gal(L/K)$. As $f$ is of degree 4 and irreducible, we have that $|Gal(L/K)| = 4$. So $Gal(L/K)$ is either isomorphic to the cyclic group (1 proper subgroup, namely $<2>$) or to the Klein-Gruppe (3 proper subgroups, namely $<(1, 0)>, <(0, 1)>, <(1, 1)>$). How do we find out which group it is isomorphic to?
For the second question I have zero clue how to solve it. I put in $\beta^k$ in $f$, but that doesn't simplify to anything nice, even knowing that $K$ has characteristic 2.