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Inmathematics, aprofinite integer is an element of thering (sometimes pronounced as zee-hat or zed-hat)

${\displaystyle \hat{\mathbb{Z}}=\underleftarrow{\lim }\mathbb{Z}/n\mathbb{Z},}$

where theinverse limit of thequotient rings${\displaystyle \mathbb{Z}/n\mathbb{Z}}$ runs through allnatural numbers${\displaystyle n}$,partially ordered bydivisibility. By definition, this ring is theprofinite completion of theintegers${\displaystyle \mathbb{Z}}$. By theChinese remainder theorem,${\displaystyle \hat{\mathbb{Z}}}$ can also be understood as thedirect product of rings

${\displaystyle \hat{\mathbb{Z}}=\prod _p{\mathbb{Z}}_p,}$

where the index${\displaystyle p}$ runs over allprime numbers, and${\displaystyle {\mathbb{Z}}_p}$ is the ring ofp-adic integers. This group is important because of its relation toGalois theory,étale homotopy theory, and the ring ofadeles. In addition, it provides a basic tractable example of aprofinite group.

The profinite integers${\displaystyle \hat{\mathbb{Z}}}$ can be constructed as the set of sequences${\displaystyle \upsilon }$ of residues represented as$${\displaystyle \upsilon =({\upsilon }_{1}mod1,{\upsilon }_{2}mod2,{\upsilon }_{3}mod3,\dots )}$$such that${\displaystyle m|n⟹{\upsilon }_m\equiv {\upsilon }_{n}modm}$.

Pointwise addition and multiplication make it a commutative ring.

The ring ofintegers embeds into the ring of profinite integers by the canonical injection:$${\displaystyle \eta :\mathbb{Z}\hookrightarrow \hat{\mathbb{Z}}}$$ where${\displaystyle n\mapsto (nmod1,nmod2,\dots ).}$It is canonical since it satisfies theuniversal property of profinite groups that, given any profinite group${\displaystyle H}$ and any group homomorphism${\displaystyle f:\mathbb{Z}\to H}$, there exists a uniquecontinuous group homomorphism${\displaystyle g:\hat{\mathbb{Z}}\to H}$ with${\displaystyle f=g\eta }$.

Every integer${\displaystyle n\ge 0}$ has a unique representation in thefactorial number system as$${\displaystyle n=\sum _{i=1}^{\mathrm{\infty }}{c}_{i}i!\text{with }{c}_i\in \mathbb{Z}}$$where${\displaystyle 0\le c_i\le i}$ for every${\displaystyle i}$, and only finitely many of${\displaystyle c_1,c_2,c_3,\dots }$ are nonzero.

Its factorial number representation can be written as${\displaystyle (\cdots c_3{c}_2{c}_1{)}_{!}}$.

In the same way, a profinite integer can be uniquely represented in the factorial number system as an infinite string${\displaystyle (\cdots c_3{c}_2{c}_1{)}_{!}}$, where each${\displaystyle c_i}$ is an integer satisfying${\displaystyle 0\le c_i\le i}$.^[1]

The digits${\displaystyle c_1,c_2,c_3,\dots ,c_{k-1}}$ determine the value of the profinite integer mod${\displaystyle k!}$. More specifically, there is a ring homomorphism${\displaystyle \hat{\mathbb{Z}}\to \mathbb{Z}/k!\mathbb{Z}}$ sending$${\displaystyle (\cdots c_3{c}_2{c}_1{)}_{!}\mapsto \sum _{i=1}^{k-1}{c}_{i}i!\mathrm{mod}k!}$$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.

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

Explicitly, the isomorphism is${\displaystyle \varphi :\prod _p{\mathbb{Z}}_p\to \hat{\mathbb{Z}}}$ by$${\displaystyle \varphi ((n_2,n_3,n_5,\cdots ))(k)=\prod _q{n}_q\mathrm{mod}k}$$where${\displaystyle q}$ ranges over all prime-power factors${\displaystyle p_i^{d_i}}$ of${\displaystyle k}$, that is,${\displaystyle k=\prod _{i=1}^l{p}_i^{d_i}}$ for some different prime numbers${\displaystyle p_1,...,p_l}$.

The set of profinite integers has an induced topology in which it is acompactHausdorff space, coming from the fact that it can be seen as a closed subset of the infinitedirect product$${\displaystyle \hat{\mathbb{Z}}\subset \prod _{n=1}^{\mathrm{\infty }}\mathbb{Z}/n\mathbb{Z}}$$which is compact with itsproduct topology byTychonoff's theorem. Note the topology on each finite group${\displaystyle \mathbb{Z}/n\mathbb{Z}}$ is given as thediscrete topology.

The topology on${\displaystyle \hat{\mathbb{Z}}}$ can be defined by the metric,^[1]$${\displaystyle d(x,y)=\frac{1}{min\{k\in {\mathbb{Z}}_{>0}:x\not\equiv ymod(k+1)!\}}}$$

Since addition of profinite integers is continuous,${\displaystyle \hat{\mathbb{Z}}}$ is a compact Hausdorffabelian group, and thus itsPontryagin dual must be a discrete abelian group.

In fact, the Pontryagin dual of${\displaystyle \hat{\mathbb{Z}}}$ is the abelian group${\displaystyle \mathbb{Q}/\mathbb{Z}}$ equipped with the discrete topology (note that it is not the subset topology inherited from${\displaystyle \mathbb{R}/\mathbb{Z}}$, which is not discrete). The Pontryagin dual is explicitly constructed by the function^[2]$${\displaystyle \mathbb{Q}/\mathbb{Z}\times \hat{\mathbb{Z}}\to U(1),(q,a)\mapsto \chi (qa)}$$where${\displaystyle \chi }$ is the character of the adele (introduced below)${\displaystyle {\mathbf{A}}_{\mathbb{Q},f}}$ induced by${\displaystyle \mathbb{Q}/\mathbb{Z}\to U(1),\alpha \mapsto e^{2\pi i\alpha }}$.^[3]

The tensor product${\displaystyle \hat{\mathbb{Z}}{\otimes }_{\mathbb{Z}}\mathbb{Q}}$ is thering of finite adeles$${\displaystyle {\mathbf{A}}_{\mathbb{Q},f}={\prod _p}^{\prime }{\mathbb{Q}}_p}$$of${\displaystyle \mathbb{Q}}$ where the symbol${\displaystyle {}^{\prime }}$ meansrestricted product. That is, an element is a sequence that is integral except at a finite number of places.^[4] There is an isomorphism$${\displaystyle {\mathbf{A}}_{\mathbb{Q}}\cong \mathbb{R}\times (\hat{\mathbb{Z}}{\otimes }_{\mathbb{Z}}\mathbb{Q})}$$

For thealgebraic closure${\displaystyle {\overline{\mathbf{F}}}_q}$ of afinite field${\displaystyle {\mathbf{F}}_q}$ of orderq, the Galois group can be computed explicitly. From the fact${\displaystyle \text{Gal}({\mathbf{F}}_{q^n}/{\mathbf{F}}_q)\cong \mathbb{Z}/n\mathbb{Z}}$ where the automorphisms are given by theFrobenius endomorphism, the Galois group of the algebraic closure of${\displaystyle {\mathbf{F}}_q}$ is given by the inverse limit of the groups${\displaystyle \mathbb{Z}/n\mathbb{Z}}$, so its Galois group is isomorphic to the group of profinite integers^[5]$${\displaystyle \mathrm{Gal}({\overline{\mathbf{F}}}_q/{\mathbf{F}}_q)\cong \hat{\mathbb{Z}}}$$which gives a computation of theabsolute Galois group of a finite field.

This construction can be re-interpreted in many ways. One of them is frométale homotopy type which defines theétale fundamental group${\displaystyle {\pi }_1^{et}(X)}$ as the profinite completion of automorphisms$${\displaystyle {\pi }_1^{et}(X)=\underset{i\in I}{lim}\text{Aut}(X_i/X)}$$where${\displaystyle X_i\to X}$ is anétale cover. Then, the profinite integers are isomorphic to the group$${\displaystyle {\pi }_1^{et}(\text{Spec}({\mathbf{F}}_q))\cong \hat{\mathbb{Z}}}$$from the earlier computation of the profinite Galois group. In addition, there is an embedding of the profinite integers inside the étale fundamental group of thealgebraic torus$${\displaystyle \hat{\mathbb{Z}}\hookrightarrow {\pi }_1^{et}({\mathbb{G}}_m)}$$since the covering maps come from thepolynomial maps$${\displaystyle (\cdot {)}^n:{\mathbb{G}}_m\to {\mathbb{G}}_m}$$from the map ofcommutative rings$${\displaystyle f:\mathbb{Z}[x,x^{-1}]\to \mathbb{Z}[x,x^{-1}]}$$ sending${\displaystyle x\mapsto x^n}$since${\displaystyle {\mathbb{G}}_m=\text{Spec}(\mathbb{Z}[x,x^{-1}])}$. If the algebraic torus is considered over a field${\displaystyle k}$, then the étale fundamental group${\displaystyle {\pi }_1^{et}({\mathbb{G}}_m/\text{Spec(k)})}$ contains an action of${\displaystyle \text{Gal}(\overline{k}/k)}$ as well from thefundamental exact sequence in étale homotopy theory.

Class field theory is a branch ofalgebraic number theory studying the abelian field extensions of a field. Given theglobal field${\displaystyle \mathbb{Q}}$, theabelianization of its absolute Galois group$${\displaystyle \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}{)}^{ab}}$$is intimately related to the associated ring of adeles${\displaystyle {\mathbb{A}}_{\mathbb{Q}}}$ and the group of profinite integers. In particular, there is a map, called theArtin map^[6]$${\displaystyle {\mathrm{\Psi }}_{\mathbb{Q}}:{\mathbb{A}}_{\mathbb{Q}}^{\times }/{\mathbb{Q}}^{\times }\to \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}{)}^{ab}}$$which is an isomorphism. This quotient can be determined explicitly as

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

• Ring of adeles
• Supernatural number

• Connes, Alain;Consani, Caterina (2015). "Geometry of the arithmetic site".arXiv:1502.05580 [math.AG].
• Milne, J.S. (2013-03-23)."Class Field Theory"(PDF). Archived fromthe original(PDF) on 2013-06-19. Retrieved2020-06-07.

• http://ncatlab.org/nlab/show/profinite+completion+of+the+integers
• https://web.archive.org/web/20150401092904/http://www.noncommutative.org/supernatural-numbers-and-adeles/
• https://euro-math-soc.eu/system/files/news/Hendrik%20Lenstra_Profinite%20number%20theory.pdf

• Algebraic number theory
• P-adic numbers
• Ring theory

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