{{Short description|Number-theoretic concept}} {{Technical|date=May 2023}} In mathematics, a '''profinite integer''' is an element of the ring (sometimes pronounced as zee-hat or zed-hat) :<math>\widehat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n\mathbb{Z},</math> where the inverse limit of the quotient rings <math>\mathbb{Z}/n\mathbb{Z}</math> runs through all natural numbers <math>n</math>, partially ordered by divisibility. By definition, this ring is the profinite completion of the integers <math>\mathbb{Z}</math>. By the Chinese remainder theorem, <math>\widehat{\mathbb{Z}}</math> can also be understood as the direct product of rings :<math>\widehat{\mathbb{Z}} = \prod_p \mathbb{Z}_p,</math> where the index <math>p</math> runs over all prime numbers, and <math>\mathbb{Z}_p</math> 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 <math>\widehat{\Z}</math> can be constructed as the set of sequences <math>\upsilon</math> of residues represented as<math display="block">\upsilon = (\upsilon_1 \bmod 1, ~ \upsilon_2 \bmod 2, ~ \upsilon_3 \bmod 3, ~ \ldots)</math>such that <math>m \ |\ n \implies \upsilon_m \equiv \upsilon_n \!\!\!\!\!\pmod{m}</math>. Pointwise addition and multiplication make it a commutative ring.
The ring of integers embeds into the ring of profinite integers by the canonical injection<math display="block">\eta: \mathbb{Z} \hookrightarrow \widehat{\mathbb{Z}},</math>where <math> n \mapsto (n \bmod 1, n \bmod 2, \dots).</math> It is canonical since it satisfies the universal property of profinite groups that, given any profinite group <math>H</math> and any group homomorphism <math>f : \Z \rightarrow H</math>, there exists a unique continuous group homomorphism <math>g : \widehat{\Z} \rightarrow H</math> with <math>f = g \eta</math>.
=== Using the factorial number system ===
Every integer <math>n \ge 0</math> has a unique representation in the factorial number system as<math display="block">n = \sum_{i=1}^\infty c_i i! \quad \text{with } c_i \in \Z,</math>where <math>0 \le c_i \le i</math> for every <math>i</math>, and only finitely many of <math>c_1,c_2,c_3,\ldots</math> are nonzero. This can be written as <math>(\cdots c_3 c_2 c_1)_!</math>.
In the same way, a profinite integer can be uniquely represented in the factorial number system as an infinite string <math>(\cdots c_3 c_2 c_1)_!</math>, where each <math>c_i</math> is an integer satisfying <math>0 \le c_i \le i</math>.<ref name="lenstra">{{cite web |last1=Lenstra |first1=Hendrik |title=Profinite number theory |url=https://www.maa.org/sites/default/files/images/mathfest/2016/pntt.pdf |website=Mathematical Association of America |access-date=11 August 2022}}</ref> The digits <math>c_1, c_2, c_3, \ldots, c_{k-1}</math> determine the value of the profinite integer modulo <math>k!</math>. More specifically, there is a ring homomorphism <math>\widehat{\Z}\to \Z / k! \, \Z</math> sending<math display="block">(\cdots c_3 c_2 c_1)_! \mapsto \sum_{i=1}^{k-1} c_i i! \bmod k!</math>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 <math>n</math> with prime factorization<math display="block">n = p_1^{a_1}\cdots p_k^{a_k}</math>of non-repeating primes, there is a ring isomorphism<math display="block">\mathbb{Z}/n \cong \mathbb{Z}/p_1^{a_1}\times \cdots \times \mathbb{Z}/p_k^{a_k}</math>from the theorem. Moreover, any surjection<math display="block">\mathbb{Z}/n \to \mathbb{Z}/m</math>will just be a map on the underlying decompositions where there are induced surjections<math display="block">\mathbb{Z}/p_i^{a_i} \to \mathbb{Z}/p_i^{b_i}</math>since we must have <math>a_i \geq b_i</math>. Under the inverse limit definition of the profinite integers, we have the isomorphism<math display="block">\widehat{\mathbb{Z}} \cong \prod_p \mathbb{Z}_p</math>with the direct product of ''p''-adic integers. Explicitly, the isomorphism is <math>\phi: \prod_p \mathbb{Z}_p \to \widehat\Z</math> by<math display="block">\phi((n_2, n_3, n_5, \cdots))(k) = \prod_{q} n_q \bmod k,</math>where <math>q</math> ranges over all prime-power factors <math>p_i^{d_i}</math> of <math>k</math>; that is, <math>k = \prod_{i=1}^l p_i^{d_i}</math> for some different prime numbers <math>p_1, ..., p_l</math>.
== Relations ==
=== Topological properties === The set of profinite integers has an induced topology in which it is a compact Hausdorff space (in fact, a Stone space) arising from the fact that it can be seen as a closed subset of the infinite direct product<math display="block">\widehat{\mathbb{Z}} \subset \prod_{n=1}^\infty \mathbb{Z}/n\mathbb{Z},</math>which is compact with its product topology by Tychonoff's theorem. The topology on each finite group <math>\mathbb{Z}/n\mathbb{Z}</math> is given as the discrete topology.
The topology on <math>\widehat{\Z}</math> can be defined by the metric<ref name="lenstra" /><math display="block">d(x,y) = \frac1{ \min\{ k \in \Z_{>0} : x \not\equiv y \bmod{(k+1)!} \} }.</math>Since addition of profinite integers is continuous, <math>\widehat{\mathbb{Z}}</math> is a compact Hausdorff abelian group, and thus its Pontryagin dual must be a discrete abelian group. In fact, the Pontryagin dual of <math>\widehat{\mathbb{Z}}</math> is the abelian group <math>\mathbb{Q}/\mathbb{Z}</math> equipped with the discrete topology (note that it is not the subset topology inherited from <math>\R/\Z</math>, which is not discrete). The Pontryagin dual is explicitly constructed by the function<ref>{{harvnb|Connes|Consani|2015|loc=§ 2.4.}}</ref><math display="block">\mathbb{Q}/\mathbb{Z} \times \widehat{\mathbb{Z}} \to U(1), \, (q, a) \mapsto \chi(qa),</math>where <math>\chi</math> is the character of the adele (introduced below) <math>\mathbf{A}_{\mathbb{Q}, f}</math> induced by <math>\mathbb{Q}/\mathbb{Z} \to U(1), \, \alpha \mapsto e^{2\pi i\alpha}</math>.<ref>K. Conrad, [http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/characterQ.pdf The character group of '''Q''']</ref>
=== Relation with adeles === The tensor product <math>\widehat{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q}</math> is the ring of finite adeles<math display="block">\mathbf{A}_{\mathbb{Q}, f} = {\prod_p}' \mathbb{Q}_p</math>of <math>\mathbb{Q}</math>, where the symbol <math>'</math> indicates a restricted product. That is, an element is a sequence that is integral except at a finite number of places.<ref>[https://math.stackexchange.com/q/233136 Questions on some maps involving rings of finite adeles and their unit groups].</ref> There is an isomorphism<math display="block">\mathbf{A}_\mathbb{Q} \cong \mathbb{R}\times(\widehat{\mathbb{Z}}\otimes_\mathbb{Z}\mathbb{Q}).</math>
=== Applications in Galois theory and étale homotopy theory === For the algebraic closure <math>\overline{\mathbf{F}}_q</math> of a finite field <math>\mathbf{F}_q</math> of order ''q,'' the Galois group can be computed explicitly. From the fact <math>\text{Gal}(\mathbf{F}_{q^n}/\mathbf{F}_q) \cong \mathbb{Z}/n\mathbb{Z}</math> where the automorphisms are given by the Frobenius endomorphism, the Galois group of the algebraic closure of <math>\mathbf{F}_q</math> is given by the inverse limit of the groups <math>\mathbb{Z}/n\mathbb{Z}</math>, so its Galois group is isomorphic to the group of profinite integers<ref>{{harvnb|Milne|2013|loc=Ch. I Example A. 5.}}</ref><math display="block">\operatorname{Gal}(\overline{\mathbf{F}}_q/\mathbf{F}_q) \cong \widehat{\mathbb{Z}},</math>which gives a computation of the absolute Galois group of a finite field.
==== Relation with étale fundamental groups of algebraic tori ==== This construction can be reinterpreted in many ways. One of them is from étale homotopy type, which defines the étale fundamental group <math>\pi_1^{et}(X)</math> as the profinite completion of automorphisms<math display="block">\pi_1^{et}(X) = \lim_{i \in I} \text{Aut}(X_i/X),</math>where <math>X_i \to X</math> is an étale cover. Then, the profinite integers are isomorphic to the group<math display="block">\pi_1^{et}(\text{Spec}(\mathbf{F}_q)) \cong \widehat{\mathbb{Z}}</math>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 the algebraic torus<math display="block">\widehat{\mathbb{Z}} \hookrightarrow \pi_1^{et}(\mathbb{G}_m),</math>since the covering maps come from the polynomial maps<math display="block">(\cdot)^n:\mathbb{G}_m \to \mathbb{G}_m</math>from the map of commutative rings<math display="block">f:\mathbb{Z}[x,x^{-1}] \to \mathbb{Z}[x,x^{-1}]</math>sending <math>x \mapsto x^n</math> since <math>\mathbb{G}_m = \text{Spec}(\mathbb{Z}[x,x^{-1}])</math>. If the algebraic torus is considered over a field <math>k</math>, then the étale fundamental group <math>\pi_1^{et}(\mathbb{G}_m/\text{Spec(k)})</math> contains an action of <math>\text{Gal}(\overline{k}/k)</math> as well from the fundamental exact sequence in étale 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 <math>\mathbb{Q}</math>, the abelianization of its absolute Galois group<math display="block">\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})^{ab}</math>is intimately related to the associated ring of adeles <math>\mathbb{A}_\mathbb{Q}</math> and the group of profinite integers. In particular, there is a map, called the Artin map<ref>{{Cite web|title=Class field theory - lccs|url=http://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html#sec13|access-date=2020-09-25|website=www.math.columbia.edu}}</ref><math display="block">\Psi_\mathbb{Q}:\mathbb{A}_\mathbb{Q}^\times / \mathbb{Q}^\times \to \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})^{ab},</math>which is an isomorphism. This quotient can be determined explicitly as<math display="block">\begin{align} \mathbb{A}_\mathbb{Q}^\times/\mathbb{Q}^\times &\cong (\mathbb{R}\times \widehat{\mathbb{Z}})/\mathbb{Z} \\ &= \underset{\leftarrow}{\lim} \mathbb({\mathbb{R}}/m\mathbb{Z}) \\ &= \underset{x \mapsto x^m}{\lim} S^1 \\ &= \widehat{\mathbb{Z}}, \end{align}</math>giving the desired relation. There is an analogous statement for local class field theory since every finite abelian extension of <math>K/\mathbb{Q}_p</math> is induced from a finite field extension <math>\mathbb{F}_{p^n}/\mathbb{F}_p</math>.
== See also == *Ring of adeles *Supernatural number
== Notes == {{reflist}}
== References == *{{cite arXiv |first1=Alain |last1=Connes |author-link=Alain Connes |first2=Caterina |last2=Consani|author2-link=Caterina Consani |eprint=1502.05580 |title=Geometry of the arithmetic site |date=2015 |class=math.AG }} *{{cite web|url=http://www.jmilne.org/math/CourseNotes/CFT.pdf |title=Class Field Theory |last=Milne |first=J.S. |date=2013-03-23 |access-date=2020-06-07 |archive-url=https://web.archive.org/web/20130619104611/http://www.jmilne.org/math/CourseNotes/CFT.pdf |archive-date=2013-06-19}}
== External links == *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
Category:Algebraic number theory Category:P-adic numbers Category:Ring theory Category:Topological groups