Profinite integer

In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat)
 * $$\widehat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n\mathbb{Z} = \prod_p \mathbb{Z}_p$$

where the inverse limit
 * $$\varprojlim \mathbb{Z}/n\mathbb{Z}$$

indicates the profinite completion of $$\mathbb{Z}$$, the index $$p$$ runs over all prime numbers, and $$\mathbb{Z}_p$$ is the ring of p-adic integers. This group is important because of its relation to Galois theory, étale homotopy theory, and the ring of adeles. In addition, it provides a basic tractable example of a profinite group.

Construction
The profinite integers $$\widehat{\Z}$$ can be constructed as the set of sequences $$\upsilon$$ of residues represented as $$\upsilon = (\upsilon_1 \bmod 1, ~ \upsilon_2 \bmod 2, ~ \upsilon_3 \bmod 3, ~ \ldots)$$ such that $$m \ |\ n \implies \upsilon_m \equiv \upsilon_n \bmod m$$.

Pointwise addition and multiplication make it a commutative ring.

The ring of integers embeds into the ring of profinite integers by the canonical injection: $$\eta: \mathbb{Z} \hookrightarrow \widehat{\mathbb{Z}}$$ where $$ n \mapsto (n \bmod 1, n \bmod 2, \dots).$$ It is canonical since it satisfies the universal property of profinite groups that, given any profinite group $$H$$ and any group homomorphism $$f : \Z \rightarrow H$$, there exists a unique continuous group homomorphism $$g : \widehat{\Z} \rightarrow H$$ with $$f = g \eta$$.

Using Factorial number system
Every integer $$n \ge 0$$ has a unique representation in the factorial number system as $$n = \sum_{i=1}^\infty c_i i! \qquad \text{with } c_i \in \Z$$ where $$0 \le c_i \le i$$ for every $$i$$, and only finitely many of $$c_1,c_2,c_3,\ldots$$ are nonzero.

Its factorial number representation can be written as $$(\cdots c_3 c_2 c_1)_!$$.

In the same way, a profinite integer can be uniquely represented in the factorial number system as an infinite string $$(\cdots c_3 c_2 c_1)_!$$, where each $$c_i$$ is an integer satisfying $$0 \le c_i \le i$$.

The digits $$c_1, c_2, c_3, \ldots, c_{k-1}$$ determine the value of the profinite integer mod $$k!$$. More specifically, there is a ring homomorphism $$\widehat{\Z}\to \Z / k! \, \Z$$ sending $$(\cdots c_3 c_2 c_1)_! \mapsto \sum_{i=1}^{k-1} c_i i! \mod k!$$ The difference of a profinite integer from an integer is that the "finitely many nonzero digits" condition is dropped, allowing for its factorial number representation to have infinitely many nonzero digits.

Using the Chinese Remainder theorem
Another way to understand the construction of the profinite integers is by using the Chinese remainder theorem. Recall that for an integer $$n$$ with prime factorization $$n = p_1^{a_1}\cdots p_k^{a_k}$$ of non-repeating primes, there is a ring isomorphism $$\mathbb{Z}/n \cong \mathbb{Z}/p_1^{a_1}\times \cdots \times \mathbb{Z}/p_k^{a_k}$$ from the theorem. Moreover, any surjection $$\mathbb{Z}/n \to \mathbb{Z}/m$$ will just be a map on the underlying decompositions where there are induced surjections $$\mathbb{Z}/p_i^{a_i} \to \mathbb{Z}/p_i^{b_i}$$ since we must have $$a_i \geq b_i$$. It should be much clearer that under the inverse limit definition of the profinite integers, we have the isomorphism $$\widehat{\mathbb{Z}} \cong \prod_p \mathbb{Z}_p$$ with the direct product of p-adic integers.

Explicitly, the isomorphism is $$\phi: \prod_p \mathbb{Z}_p \to \widehat\Z$$ by $$\phi((n_2, n_3, n_5, \cdots))(k) = \prod_{q} n_q \mod k$$ where $$q$$ ranges over all prime-power factors $$p_i^{d_i}$$ of $$k$$, that is, $$k = \prod_{i=1}^l p_i^{d_i}$$ for some different prime numbers $$p_1, ..., p_l$$.

Topological properties
The set of profinite integers has an induced topology in which it is a compact Hausdorff space, coming from the fact that it can be seen as a closed subset of the infinite direct product $$\widehat{\mathbb{Z}} \subset \prod_{n=1}^\infty \mathbb{Z}/n\mathbb{Z}$$ which is compact with its product topology by Tychonoff's theorem. Note the topology on each finite group $$\mathbb{Z}/n\mathbb{Z}$$ is given as the discrete topology.

The topology on $$\widehat{\Z}$$ can be defined by the metric, $$d(x,y) = \frac1{ \min\{ k \in \Z_{>0} : x \not\equiv y \bmod{(k+1)!} \} }$$

Since addition of profinite integers is continuous, $$\widehat{\mathbb{Z}}$$ is a compact Hausdorff abelian group, and thus its Pontryagin dual must be a discrete abelian group.

In fact, the Pontryagin dual of $$\widehat{\mathbb{Z}}$$ is the abelian group $$\mathbb{Q}/\mathbb{Z}$$ equipped with the discrete topology (note that it is not the subset topology inherited from $$\R/\Z$$, which is not discrete). The Pontryagin dual is explicitly constructed by the function $$\mathbb{Q}/\mathbb{Z} \times \widehat{\mathbb{Z}} \to U(1), \, (q, a) \mapsto \chi(qa)$$ where $$\chi$$ is the character of the adele (introduced below) $$\mathbf{A}_{\mathbb{Q}, f}$$ induced by $$\mathbb{Q}/\mathbb{Z} \to U(1), \, \alpha \mapsto e^{2\pi i\alpha}$$.

Relation with adeles
The tensor product $$\widehat{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q}$$ is the ring of finite adeles $$\mathbf{A}_{\mathbb{Q}, f} = {\prod_p}' \mathbb{Q}_p$$ of $$\mathbb{Q}$$ where the symbol $$'$$ means restricted product. That is, an element is a sequence that is integral except at a finite number of places. There is an isomorphism $$\mathbf{A}_\mathbb{Q} \cong \mathbb{R}\times(\hat{\mathbb{Z}}\otimes_\mathbb{Z}\mathbb{Q})$$

Applications in Galois theory and Etale homotopy theory
For the algebraic closure $$\overline{\mathbf{F}}_q$$ of a finite field $$\mathbf{F}_q$$ of order q, the Galois group can be computed explicitly. From the fact $$\text{Gal}(\mathbf{F}_{q^n}/\mathbf{F}_q) \cong \mathbb{Z}/n\mathbb{Z}$$ where the automorphisms are given by the Frobenius endomorphism, the Galois group of the algebraic closure of $$\mathbf{F}_q$$ is given by the inverse limit of the groups $$\mathbb{Z}/n\mathbb{Z}$$, so its Galois group is isomorphic to the group of profinite integers $$\operatorname{Gal}(\overline{\mathbf{F}}_q/\mathbf{F}_q) \cong \widehat{\mathbb{Z}}$$ which gives a computation of the absolute Galois group of a finite field.

Relation with Etale fundamental groups of algebraic tori
This construction can be re-interpreted in many ways. One of them is from Etale homotopy theory which defines the Etale fundamental group $$\pi_1^{et}(X)$$ as the profinite completion of automorphisms $$\pi_1^{et}(X) = \lim_{i \in I} \text{Aut}(X_i/X)$$ where $$X_i \to X$$ is an Etale cover. Then, the profinite integers are isomorphic to the group $$\pi_1^{et}(\text{Spec}(\mathbf{F}_q)) \cong \hat{\mathbb{Z}}$$ from the earlier computation of the profinite Galois group. In addition, there is an embedding of the profinite integers inside the Etale fundamental group of the algebraic torus $$\hat{\mathbb{Z}} \hookrightarrow \pi_1^{et}(\mathbb{G}_m)$$ since the covering maps come from the polynomial maps $$(\cdot)^n:\mathbb{G}_m \to \mathbb{G}_m$$ from the map of commutative rings $$f:\mathbb{Z}[x,x^{-1}] \to \mathbb{Z}[x,x^{-1}]$$ sending $$x \mapsto x^n$$ since $$\mathbb{G}_m = \text{Spec}(\mathbb{Z}[x,x^{-1}])$$. If the algebraic torus is considered over a field $$k$$, then the Etale fundamental group $$\pi_1^{et}(\mathbb{G}_m/\text{Spec(k)})$$ contains an action of $$\text{Gal}(\overline{k}/k)$$ as well from the fundamental exact sequence in etale homotopy theory.

Class field theory and the profinite integers
Class field theory is a branch of algebraic number theory studying the abelian field extensions of a field. Given the global field $$\mathbb{Q}$$, the abelianization of its absolute Galois group $$\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})^{ab}$$ is intimately related to the associated ring of adeles $$\mathbb{A}_\mathbb{Q}$$ and the group of profinite integers. In particular, there is a map, called the Artin map $$\Psi_\mathbb{Q}:\mathbb{A}_\mathbb{Q}^\times / \mathbb{Q}^\times \to \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})^{ab}$$ which is an isomorphism. This quotient can be determined explicitly as

$$\begin{align} \mathbb{A}_\mathbb{Q}^\times/\mathbb{Q}^\times &\cong (\mathbb{R}\times \hat{\mathbb{Z}})/\mathbb{Z} \\ &= \underset{\leftarrow}{\lim} \mathbb({\mathbb{R}}/m\mathbb{Z}) \\ &= \underset{x \mapsto x^m}{\lim} S^1 \\ &= \hat{\mathbb{Z}} \end{align}$$

giving the desired relation. There is an analogous statement for local class field theory since every finite abelian extension of $$K/\mathbb{Q}_p$$ is induced from a finite field extension $$\mathbb{F}_{p^n}/\mathbb{F}_p$$.