Does there exist two different fields over $F$ such that they have the same intermediate fields?
Suppose $K_1$ and $K_2$ are two different field extesnions over field $F$, and all the proper intermediate fields in the field extension $K_1$ of $F$ and $K_2$ of $F$ are the same.(here we suppose both of the field extensions have more than $1$ intermidiate field, i.e. $F$ is not the only intermediate field of the extensions above), can we find such fields $K_1,K_2$ and $F$?
Consider $F=\Bbb Q$, $K_1=\Bbb Q(\sqrt[4]{2})$ and $K_2=\Bbb Q(\sqrt{2+\sqrt{2}})$. In both cases, the only intermediate field is $\Bbb Q(\sqrt{2})$. (Besides $\Bbb Q$, of course)