# How many intermediate fields?

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?

• The claim is false as stated. In this answer I give an example of a degree four extension with infinitely many intermediate fields. – Jyrki Lahtonen Aug 14 at 7:54

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 $$G\setminus H$$.