This question grew out of the following problem.
Prove that if $L/K$ is a separable extension of degree $4$, then there are at most $3$ intermediate extensions.
I was wondering if this separable condition was necessary or if it was inserted to make the question simpler.
Using the only example of a inseparable extension I knew (i.e. extensions of rational function field), I was thinking of classifying the intermediate extensions of $L/K$ where $L=\mathbb{F}_2(\sqrt{x},\sqrt{y})$ and $K=\mathbb{F}_2(x,y)$.
I came to the conclusion that this degree 4 extension had more than $3$ intermediate fields. Apart from $F_1=K(\sqrt{x}), F_2=K(\sqrt{y}), F_3=K(\sqrt{xy})$, $F_4=K(\sqrt{x}+\sqrt{y})$ was a different intermediate extension.
(Compare this to the case when the characteristic of the field is not $2$, for example, $K=\mathbb{Q}(x,y)$ and $L=K(\sqrt{x},\sqrt{y})$. Then, $F=K(\sqrt{x}+\sqrt{y})=L$ by using the trick $\sqrt{x}=\frac{1}{2}(\sqrt{x}+\sqrt{y})+\frac{1}{2}(\sqrt{x}-\sqrt{y})=\frac{1}{2}(\sqrt{x}+\sqrt{y})+\frac{1}{2}\frac{x-y}{(\sqrt{x}+\sqrt{y})}$)
My question is my reasoning correct?
Moreover, can I classify all intermediate extensions of this extension $\mathbb{F}_2(\sqrt{x},\sqrt{y})/\mathbb{F}_2(x,y)$? (If I am not mistaken $K(\sqrt{x^i},K(\sqrt{y^j}, K(\sqrt{x^i}+\sqrt{y^j})$ give us distinct intermediate extensions for odd $i,j$ but there are many more. Is there a simple way to list all such extensions?)
If you have any reference for intermediate field extensions for inseparable extensions, that would also be very nice.