Let $F$ be a finite field. There is an isomorphism of topological groups $\left(\mathrm{Gal}(\overline{F}/F),\circ\right) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the structure of a topological *ring* isomorphic to $\widehat{\mathbb{Z}}$. What does the multiplication $*$ look like? If $\sigma$ is the Frobenius, we have $\sigma^n * \sigma^m = \sigma^{n*m}$, and this describes $*$ completely. Is there any way to give an explicit and *natural* formula for $\alpha * \beta$ if $\alpha,\beta$ are $F$-automorphisms of $\overline{F}$? 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 $\mathcal{C}$ of finite étale $F$-algebras together with the fiber functor to $\mathsf{FinSet}$. The automorphism group is exactly $\pi_1(\mathrm{Spec}(F))=\widehat{\mathbb{Z}}$. Which additional structure on the Galois category $\mathcal{C}$ is responsible for the ring structure on its automorphism group?