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Profinite Integers
In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) :\widehat = \varprojlim \mathbb/n\mathbb = \prod_p \mathbb_p where :\varprojlim \mathbb/n\mathbb indicates the profinite completion of \mathbb, the index p runs over all prime numbers, and \mathbb_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 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 \hookr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other top ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Étale Fundamental Group
The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces. Topological analogue/informal discussion In algebraic topology, the fundamental group \pi_1(X,x) of a pointed topological space (X, x) is defined as the group of homotopy classes of loops based at x. This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the Zariski topology. In the classification of covering spaces, it is shown that the fundamental group is exactly the group of deck transformations of the universal covering space. This is more promising: finite étale morphisms of algebraic varieties are the appropriate analogue of covering spaces of topological spaces. Unfortunately, an algebraic variety X often fails to have a "universal cover" that is finite over ''X'', so one must consider the entire category of finite étale coverings of ''X''. O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Étale Homotopy Type
In mathematics, especially in algebraic geometry, the étale homotopy type is an analogue of the homotopy type of topological spaces for algebraic varieties. Roughly speaking, for a variety or scheme ''X'', the idea is to consider étale coverings U \rightarrow X and to replace each connected component of ''U'' and the higher "intersections", i.e., fiber products, U_n := U \times_X U \times_X \dots \times_X U (''n''+1 copies of ''U'', n \geq 0) by a single point. This gives a simplicial set which captures some information related to ''X'' and the étale topology of it. Slightly more precisely, it is in general necessary to work with étale hypercovers (U_n)_ instead of the above simplicial scheme determined by a usual étale cover. Taking finer and finer hypercoverings (which is technically accomplished by working with the pro-object in simplicial sets determined by taking all hypercoverings), the resulting object is the étale homotopy type of ''X''. Similarly to classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Galois Group
In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' that fix ''K''. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group. (When ''K'' is a perfect field, ''K''sep is the same as an algebraic closure ''K''alg of ''K''. This holds e.g. for ''K'' of characteristic zero, or ''K'' a finite field.) Examples * The absolute Galois group of an algebraically closed field is trivial. * The absolute Galois group of the real numbers is a cyclic group of two elements (complex conjugation and the identity map), since C is the separable closure of R and ''C:Rnbsp;= 2. * The absolute Galois group of a finite field ''K'' is isomorphic to the group :: \hat = \varprojlim \mathbf/n\mathbf. (For the notation, see Inverse limit.) :The Frobenius automorphis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Endomorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism maps every element to its -th power. In certain contexts it is an automorphism, but this is not true in general. Definition Let be a commutative ring with prime characteristic (an integral domain of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism ''F'' is defined by :F(r) = r^p for all ''r'' in ''R''. It respects the multiplication of ''R'': :F(rs) = (rs)^p = r^ps^p = F(r)F(s), and is 1 as well. Moreover, it also respects the addition of . The expression can be expanded using the binomial theorem. Because is prime, it divides but not any for ; it therefore will divide the numerator, but not the denominator, of the explicit formula of the binomial coefficients :\frac, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky (1972) pp.74-76 or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field ''K'' is unique up to an isomorphism that fixes every member of ''K''. Because of this essential uniqueness, we often speak of ''the'' algebraic closure of ''K'', rather than ''an'' algebraic closure of ''K''. The algebraic closure of a field ''K'' can be thought of as the largest algebraic extension of ''K''. To see this, note that if ''L'' is any algebraic extension of ''K'', then the algebraic closure of ''L'' is also an algebraic closure of ''K'', and so ''L'' is contained within the algebraic closure of ''K''. The algebraic closure of ''K'' is also the smallest algebraically closed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Restricted Product
In mathematics, the restricted product is a construction in the theory of topological groups. Let I be an index set; S a finite subset of I. If G_i is a locally compact group for each i \in I, and K_i \subset G_i is an open compact subgroup for each i \in I \setminus S, then the restricted product : \prod_i\nolimits' G_i\, is the subset of the product of the G_i 's consisting of all elements (g_i)_ such that g_i \in K_i for all but finitely many i \in I \setminus S. This group is given the topology whose basis of open sets are those of the form : \prod_i A_i\,, where A_i is open in G_i and A_i = K_i for all but finitely many i. One can easily prove that the restricted product is itself a locally compact group. The best known example of this construction is that of the adele ring and idele group of a global field. See also *Direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogousl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Finite Adeles
Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and literature * ''The Ring'' (franchise), a Japanese horror media franchise based on the novel series by Koji Suzuki ** ''Ring'' (novel series) *** ''Ring'' (Suzuki novel), 1991 ** ''Ring'' (film), or ''The Ring'', a 1998 Japanese horror film by Hideo Nakata *** ''The Ring'' (2002 film), an American horror film, remake of the 1998 Japanese film ** ''Ring'' (1995 film), a TV film ** ''Rings'' (2005 film), a short film by Jonathan Liebesman ** ''Rings'' (2017 film), an American horror film * ''Ring'' (Baxter novel), a 1994 science fiction novel * ''Ring'' (Alexis novel), a 2021 Canadian novel by André Alexis Gaming * ''Ring'' (video game), 1998 * Rings (''Sonic the Hedgehog''), a collectible in ''Sonic the Hedgehog'' games Music ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pontryagin Dual
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and the additive group of the integers (also with the discrete topology), the real numbers, and every finite dimensional vector space over the reals or a -adic field. The Pontryagin dual of a locally compact abelian group is the locally compact abelian topological group formed by the continuous group homomorphisms from the group to the circle group with the operation of pointwise multiplication and the topology of uniform convergence on compact sets. The Pontryagin duality theorem establishes Pontryagin duality by stating that any locally compact abelian group is naturally isomorphic with its bidual (the dual of its dual). The Fourier inversion theorem is a special case of this the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology. Definitions Given a set X: A metric space (E,d) is said to be '' uniformly discrete'' if there exists a ' r > 0 such that, for any x,y \in E, one has either x = y or d(x,y) > r. The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set \left\. Properties The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
