Profinite integer

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 \!\!\!\!\!\pmod{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 the factorial number system Every integer n \ge 0 has a unique representation in the factorial number system asn = \sum_{i=1}^\infty c_i i! \quad \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. This 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 modulo 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! \bmod 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 factorizationn = 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}/mwill 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. Under the inverse limit definition of the profinite integers, we have the isomorphism\widehat{\mathbb{Z}} \cong \prod_p \mathbb{Z}_pwith 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 \bmod 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. == 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\widehat{\mathbb{Z}} \subset \prod_{n=1}^\infty \mathbb{Z}/n\mathbb{Z},which is compact with its product topology by Tychonoff's theorem. 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\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}_pof \mathbb{Q}, where the symbol ' indicates a 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(\widehat{\mathbb{Z}}\otimes_\mathbb{Z}\mathbb{Q}). Applications in Galois theory and étale 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 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 \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 étale cover. Then, the profinite integers are isomorphic to the group\pi_1^{et}(\text{Spec}(\mathbf{F}_q)) \cong \widehat{\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 the algebraic torus\widehat{\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}_mfrom the map of commutative ringsf:\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 étale 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 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 \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 \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}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. == See also ==