Consider F―p, the algebraic closure of Fp. I want to see that: for every proper subfield K≤Fp―,F―pK is not a finite extension. It is known that, and can be somewhat easily shown that F―p=∪n≥1Fpn. Now, if any of the proper subfields have the form Fpn, it is easy enough to see that F―p≠Fpn(α1,⋯,αm) for some αi by going high up enough, i.e, to some big enough m such that αi∈Fpm⊆Fp― The problem is characterizing the proper subfields. Is every subfield of Fp― going to have this form? Can we have an infinite intermediate subfield?

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Consider F―p, the algebraic closure of Fp. I want to see that: for every proper subfield K≤Fp―,F―pK is not a finite extension.   It is known that, and can be somewhat easily shown that F―p=∪n≥1Fpn.  Now, if any of the proper subfields have the form Fpn, it is easy enough to see that F―p≠Fpn(α1,⋯,αm) for some αi by going high up enough, i.e, to some big enough m such that αi∈Fpm⊆Fp―  The problem is characterizing the proper subfields. Is every subfield of Fp― going to have this form? Can we have an infinite intermediate subfield?

Annie Levasseur · Accepted Answer

Step 1As Slade explained (+1) this follows either from the known structure of the automorphism group of Fp― or from a theorem of Artin &amp;amp; Schreier stating that any finite extension K―K, with the bigger field algebraically closed, is of the form K―=K(−1).Step 2An elementary argument can also be given. Assume that Fp―K is a finite extension. Then that extension is Galois. It is obviously normal, and it is also separable because every element of Fp― belongs to a finite field, and hence is a zero of a separable polynomial over the &#039; field (and hence also over K). By basic Galois theory this implies that Fp― has an automorphism of a finite order.

Daniel Cormack · Accepted Answer

The Galois group here is isomorphic to G=Z^, the profinite integers. Since G is torsion-free, there are no finite index subfields.This also follow from Artin-Schreier, though I don&#039;t know offhand of an extremely clean and short proof in the case of finite fields.

Consider \overline{F}_{p}, the algebraic closure of F_{p}. I want to see that: for every

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