# Finite separable extension of fields imply the number of intermediate subfield is finite

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.

• 1. To prove this theorem in the etale site one needs some osrt of finiteness criterion on the base, I think. Imagine the base an infintie union of points, and the cover a 2-to-1 cover at each point. Then there are infinitely many subcovers, corresponding to which points are 2-to-1 and which are 1-to-1. But even a connectedness condition deals with this problem. 2. Have you tried to figure out a proof in a Grothendieck's Galois Theory direction? I'm not sure what geometric facts about finite etale covers one needs to show they form a Galois category, but once you do you get finiteness. – Will Sawin Feb 18 '14 at 2:21
• The characterization of when a finite extension (separable or not) admits finitely many intermediate fields is exactly that it is a primitive extension. This is all far more elementary than any categorical formalism. – user76758 Feb 18 '14 at 2:53
• Can't you use that separable means that tensoring with the algebraic closure is a finite direct product of copies of the algebraic closure? This has finitely many subalgebras. Tensoring with the algebraic closure doesn't identify any intermediate fields and so you had finitely many to start with. (I'm a bit sleepy so this might be wrong.) – Benjamin Steinberg Feb 18 '14 at 3:56