# Ring structure on the Galois group of a finite field

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?

Yes, there is a natural way of describing the ring structure on $Gal(\overline{F}\backslash F)$, 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 $F=\mathbb{F}_p$.

For each $n\in\mathbb{N}$, $Gal(\mathbb{F}_{p^n}\backslash\mathbb{F}_p)$ is cyclic of order $n$, generated by the Frobenius $\sigma(x)=x^p$, so there are natural group isomorphisms $\iota_n:\frac{\mathbb{Z}}{n\mathbb{Z}}\cong Gal(\mathbb{F}_{p^n}\backslash\mathbb{F}_p), r+n\mathbb{Z}\mapsto \sigma^r$.

Therefore $Gal(\mathbb{F}_{p^m}\backslash\mathbb{F}_p)$ carries a natural ring structure given by $\sigma^r*\sigma^s=\sigma^{rs}$, and this makes the maps $\iota_n$ ring isomorphisms.

There are natural surjective transition maps $\nu_{n,m}:Gal(\mathbb{F}_{p^n}\backslash\mathbb{F}_p)\to Gal(\mathbb{F}_{p^m}\backslash\mathbb{F}_p), \alpha\to\alpha|_{\mathbb{F}_{p^m}}$ whenever $m\mid n$, making this collection of Galois groups into an inverse system, and it is a fact in Galois theory that $Gal(\overline{F}\backslash\mathbb{F}_p)$ is the limit of this system.

Therefore, given general $F$-automorphisms $\alpha,\beta\in Gal(\overline{F}\backslash\mathbb{F}_p)$, we can write $\alpha=(\alpha_n)_{n\in\mathbb{N}}$, $\beta=(\beta_n)_{n\in\mathbb{N}}$, where $\alpha_n,\beta_n\in Gal(\mathbb{F}_{p^n}\backslash\mathbb{F}_p)$, $\nu_{n,m}(\alpha_n)=\alpha_m$, $\nu_{n,m}(\beta_n)=\beta_m$, and multiplication in $Gal(\overline{F}\backslash\mathbb{F}_p)$ is given by $\alpha*\beta=(\alpha_n*\beta_n)_{n\in\mathbb{N}}$, where multiplication in $Gal(\mathbb{F}_{p^n}\backslash\mathbb{F}_p)$ was defined above.

This is the natural way of describing the ring structure on $Gal(\overline{F}\backslash\mathbb{F}_p)$, and since $\widehat{\mathbb{Z}}$ is identivcally defined as the inverse limit of the system of cyclic groups $\frac{\mathbb{Z}}{n\mathbb{Z}}$, it is equivalent to the ring structure on $\widehat{\mathbb{Z}}$.