If I have a separable finite extension $L/K$, is it true that there are finitely many intermediate subfields? I know this is true if it is also a normal extension by the fundamental theorem of Galois theory. I can also show this if I use the primitive element theorem however I'm trying to use this fact to then prove the primitive element theorem later.
What I want to do is try to make $L$ sit inside some normal extension of $K$ but is that possible?
Thanks
Best Answer
Extend $L$ to its Galois closure $M$ and let $G$ be its Galois group. Any intermediate extension between $K$ and $L$ is also an intermediate extension between $K$ and $M$ and these are in bijection with the subgroups of $G$. $G$ is a finite group, so its subgroups are finite in number.
More precisely, let $L$ be the field of elements fixed by the subgroup $H$ of $G$. The subextensions of $L$ are the fields of invariants of the subgroups of $G$ which contain $H$.