Let F be a finite field. There is an isomorphism of topological groups (Gal(¯F/F),∘)≅(ˆZ,+). It follows that the Galois group carries the structure of a topological ring isomorphic to ˆZ. What does the multiplication ∗ look like? If σ is the Frobenius, we have σn∗σm=σn∗m, and this describes ∗ completely. Is there any way to give an explicit and natural formula for α∗β if α,β are F-automorphisms of ¯F? Also, is there any more conceptual reason why the Galois group carries the structure of a topological ring (without computing the Galois group)?
Maybe the following is a more precise version of the question: Consider the Galois category C of finite étale F-algebras together with the fiber functor to FinSet. The automorphism group is exactly π1(Spec(F))=ˆZ. Which additional structure on the Galois category C is responsible for the ring structure on its automorphism group?
Yes, there is a natural way of describing the ring structure on Gal(¯F∖F), just using the Galois theoretic fact that Galois groups of infinite Galois extensions can be considered as inverse limits of systems of Galois groups of finite extensions.
For simplicity, suppose that F=Fp.
For each n∈N, Gal(Fpn∖Fp) is cyclic of order n, generated by the Frobenius σ(x)=xp, so there are natural group isomorphisms ιn:ZnZ≅Gal(Fpn∖Fp),r+nZ↦σr.
Therefore Gal(Fpm∖Fp) carries a natural ring structure given by σr∗σs=σrs, and this makes the maps ιn ring isomorphisms.
There are natural surjective transition maps νn,m:Gal(Fpn∖Fp)→Gal(Fpm∖Fp),α→α|Fpm whenever m∣n, making this collection of Galois groups into an inverse system, and it is a fact in Galois theory that Gal(¯F∖Fp) is the limit of this system.
Therefore, given general F-automorphisms α,β∈Gal(¯F∖Fp), we can write α=(αn)n∈N, β=(βn)n∈N, where αn,βn∈Gal(Fpn∖Fp), νn,m(αn)=αm, νn,m(βn)=βm, and multiplication in Gal(¯F∖Fp) is given by α∗β=(αn∗βn)n∈N, where multiplication in Gal(Fpn∖Fp) was defined above.
This is the natural way of describing the ring structure on Gal(¯F∖Fp), and since ˆZ is identivcally defined as the inverse limit of the system of cyclic groups ZnZ, it is equivalent to the ring structure on ˆZ.