How many intermediate fields?

Question

Let $F ⊂ L $ a extension of fields of degree 4. Prove that there are no more than 3 fields proper intermediate subfields $K$; namely, such that $F ⊂ K ⊂ L$

Using the degree of the field extension, I only know that $K$ is a field extension of $F$ of degree 2. This question is quite more specific than related questions on stack already. So how can we solve this?

Answer 1

With the extra assumption that $L/F$ is separable (for example when characteristic zero, or when both are finite) this can be seen as follows. Justify the claims/steps:

1. There exists a field $M$, $F\subset L\subseteq M$, such that $M/F$ is Galois.
2. Let $G=Gal(M/F)$ and let $H=Gal(M/L)$ be the subgroup corresponding to $L$. Then $[G:H]=4$.
3. The intermediate fields $K$ correspond to intermediate groups $H'$ such that $H\subset H'\subset G$. We have $[H':H]=2$ for all such subgroups $H'$.
4. $H'$ is the union $H$ and a coset $gH$ for some $g\in G\setminus H$. The choice of the coset $gH$ determines $H'$ uniquely.
5. There are three cosets of $H$ in $G\setminus H$.