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1. What the question is asking

Let (F) be a finite field, (\overline{F}) a separable closure and
(G=\operatorname{Gal}(\overline{F}/F)) its absolute Galois group.

  1. We know that (G) is (as a topological group) isomorphic to the profinite completion (\widehat{\mathbb Z}).
    Via this identification one can declare a multiplication on (G) by transporting the usual multiplication of (\widehat{\mathbb Z}): if (\sigma,\tau\in G) correspond to (\hat n,\hat m\in\widehat{\mathbb Z}) then (\sigma*\tau) is the element that corresponds to (\hat n\hat m).

  2. The problem asks for an intrinsic description of the product () that does not appeal to the explicit isomorphism (G\cong\widehat{\mathbb Z}).
    In other words: given two automorphisms (\sigma,\tau) of (\overline{F}) fixing (F), can we write down (\sigma    \tau) using only the structure of the field extensions, the Frobenius map, etc.?

  3. Moreover, it asks why a ring structure appears at all: which extra data on the Galois (or étale) category of (F) forces its fundamental group to be a topological ring rather than merely a profinite group?

The answer below gives

  • an intrinsic definition of the product (*);
  • a conceptual explanation in terms of the canonical Frobenius endomorphism of the étale site (or of the fiber functor);
  • a short list of typical pitfalls when one works with this situation.

2. The canonical Frobenius and the intrinsic description of (*)

2.1 The arithmetic Frobenius is canonical

Let (|F|=q) ((q=p^{r}) with (p) prime).
For every finite étale (F)-algebra (A) there is a distinguished (F)-endomorphism [ \Phi_{A}:A\longrightarrow A,\qquad a\mapsto a^{q}. ] When (A) is a finite field extension (L/F) the map (\Phi_{L}) is the usual arithmetic Frobenius (it sends (x) to (x^{q})).

If we view a finite étale (F)-algebra as a finite set equipped with a continuous action of the absolute Galois group, (\Phi_{A}) is exactly the action of a single element (\operatorname{Fr}_{F}\in G), the Frobenius automorphism of (\overline{F}) :

[ \operatorname{Fr}_{F}(\alpha)=\alpha^{q}\qquad(\alpha\in\overline{F}). ]

The element (\operatorname{Fr}_{F}) is characterised intrinsically as the unique element of (G) whose restriction to any finite Galois extension (L/F) coincides with the arithmetic Frobenius of (L). Thus the generator of the pro‑cyclic group (G) does not depend on any choice of isomorphism with (\widehat{\mathbb Z}).

2.2 Exponents in (\widehat{\mathbb Z}) are defined by restriction

For any (\sigma\in G) and any finite Galois extension (L/F) we have a restriction homomorphism [ \operatorname{res}{L}\colon G\longrightarrow \operatorname{Gal}(L/F). ] Since each (\operatorname{Gal}(L/F)) is a *finite cyclic* group generated by (\operatorname{Fr}{L}:=\operatorname{Fr}{F}|{L}), there exists a unique element (e_{L}(\sigma)\in\mathbb Z/| \operatorname{Gal}(L/F) |\mathbb Z) such that [ \operatorname{res}{L}(\sigma)=\operatorname{Fr}{L}^{\,e_{L}(\sigma)} . \tag{1} ]

The family ({e_{L}(\sigma)}{L}) is compatible when (L\subseteq L’) (because the restriction maps commute with powers of Frobenius).
Therefore the compatible system determines a unique element [ e(\sigma)\in\widehat{\mathbb Z}=\varprojlim{L}\mathbb Z/| \operatorname{Gal}(L/F) |\mathbb Z ] such that (e_{L}(\sigma)) is the image of (e(\sigma)) in the finite quotient. In other words, (e(\sigma)) is the exponent of (\sigma) with respect to the canonical generator (\operatorname{Fr}_{F}).

The map [ e\colon G\longrightarrow\widehat{\mathbb Z},\qquad \sigma\mapsto e(\sigma) ] is a continuous isomorphism of topological groups; it is characterised by the single condition [ e(\operatorname{Fr}_{F})=1. \tag{2} ]

2.3 Intrinsic definition of the product

Now we can define the product (*) on (G) without mentioning the identification (G\cong\widehat{\mathbb Z}) at the outset.

Definition.
For (\sigma,\tau\in G) let (e(\sigma),e(\tau)\in\widehat{\mathbb Z}) be their exponents defined above. Set [ \boxed{\;\sigma * \tau\;:=\;\operatorname{Fr}_{F}^{\,e(\sigma)\,e(\tau)}\;}. \tag{3} ]

Because the exponent map (e) is a group isomorphism, (3) makes (G) into a ring whose additive structure is the original group law and whose multiplication satisfies [ \operatorname{Fr}{F}^{\,a} * \operatorname{Fr}{F}^{\,b} =\operatorname{Fr}_{F}^{\,ab}\qquad (a,b\in\widehat{\mathbb Z}), ] exactly the rule that was used in the original “computational’’ description. No external isomorphism has been used: the only data employed are

  • the canonical Frobenius element (\operatorname{Fr}_{F}), and
  • the fact that every finite quotient of (G) is cyclic generated by the restriction of (\operatorname{Fr}_{F}).

Thus (3) is an intrinsic formula for (*).

3. Why a ring structure appears: the categorical point of view

3.1 The fiber functor and its natural endomorphisms

Let (\mathcal C) be the Galois category of finite étale (F)-algebras. Fix the usual fiber functor [ \omega\colon \mathcal C \longrightarrow \mathbf{FinSet},\qquad A\longmapsto \operatorname{Hom}_{F}(A,\overline{F}). ]

For each integer (n\ge 0) the map [ \Phi^{(n)}{A}\colon\omega(A)\longrightarrow\omega(A),\qquad \varphi\mapsto\varphi^{\,q^{\,n}} ] is natural in (A); the collection ({\Phi^{(n)}}{n\in\mathbb Z}) gives a continuous homomorphism \

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