Consider , the algebraic closure of . I want to see that: for every proper subfield is not a finite extension.
It is known that, and can be somewhat easily shown that .
Now, if any of the proper subfields have the form , it is easy enough to see that for some by going high up enough, i.e, to some big enough m such that
The problem is characterizing the proper subfields. Is every subfield of going to have this form? Can we have an infinite intermediate subfield?