The proof of statements either uses Galois theory or Artin primitive element theorem.I would like to know whether there is a proof without using these.The reason to avoid using Galois theory is that we need to take normal closure to get Galois extension and then use fundamental theorem to derive the result which seems not very straightforward.
In fact,what I really would like to know is that how to characterize the "number of intermediate field is finite" in some general nonsense way.I wonder whether there exists certain "proof" or just "explanations" going as follows:
"finite separable field extension" correspondence to "etale morphism" between two schemes and etale morphisms=formally etale +locally finite presentable.I wonder how this finiteness condition will imply the "finiteness condition" of the number of intermediate fields.
I was trying to use category of subextensions of $k\rightarrow F$ and trying to say that objet $k\rightarrow F$ is object of finite type (or noetherian object) in this category.But then I realized that the number of intermediate fields is finite is much stronger than what I was trying to show. (actually,we just need finite degree extension to show that).I dont understand how the finiteness conditions contained in separable will imply the results.
Notice that,in the proof using Galois theory,one uses that Galois group is finite,then number of subgroups is finite then correspondence intermediate subfield is also finite.It is still kind of re interpret finiteness condition again and again.
