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Steinberg Symbol
In mathematics a Steinberg symbol is a pairing function which generalises the Hilbert symbol and plays a role in the algebraic K-theory of fields. It is named after mathematician Robert Steinberg. For a field ''F'' we define a ''Steinberg symbol'' (or simply a ''symbol'') to be a function ( \cdot , \cdot ) : F^* \times F^* \rightarrow G, where ''G'' is an abelian group, written multiplicatively, such that * ( \cdot , \cdot ) is bimultiplicative; * if a+b = 1 then (a,b) = 1. The symbols on ''F'' derive from a "universal" symbol, which may be regarded as taking values in F^* \otimes F^* / \langle a \otimes 1-a \rangle. By a theorem of Matsumoto, this group is K_2 F and is part of the Milnor K-theory for a field. Properties If (⋅,⋅) is a symbol then (assuming all terms are defined) * (a, -a) = 1 ; * (b, a) = (a, b)^ ; * (a, a) = (a, -1) is an element of order 1 or 2; * (a, b) = (a+b, -b/a) . Examples * The trivial symbol which is identically 1. * The Hilbert symbol on ''F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Symbol
In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from ''K''× × ''K''× to the group of ''n''th roots of unity in a local field ''K'' such as the fields of reals or p-adic numbers . It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields. The Hilbert symbol has been generalized to higher local fields. Quadratic Hilbert symbol Over a local field ''K'' whose multiplicative group of non-zero elements is ''K''×, the quadratic Hilbert symbol is the function (–, –) from ''K''× × ''K''× to defined by :(a,b)=\begin+1,&\mboxz^2=ax^2+by^2\mbox(x,y,z)\in K^3;\\-1,&\mbox\end Equivalently, (a, b) = 1 if and only if b is equal to the norm of an element of the quadratic extension Ksqrt/math> page 110. Properties The follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups have been studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis. Formal definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, ''The Daily Princetonian'', and later added book publishing to it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Studies In Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats. List of books *1 ''The General Topology of Dynamical Systems'', Ethan Akin (1993, ) *2 ''Combinatorial Rigidity'', Jack Graver, Brigitte Servatius, Herman Servatius (1993, ) *3 ''An Introduction to Gröbner Bases'', William W. Adams, Philippe Loustaunau (1994, ) *4 ''The Integrals of Lebesgue, Denjoy, Perron, and Henstock'', Russell A. Gordon (1994, ) *5 ''Algebraic Curves and Riemann Surfaces'', Rick Miranda (1995, ) *6 ''Lectures on Quantum Groups'', Jens Carsten Jantzen (1996, ) *7 ''Algebraic Number Fields'', Gerald J. Janusz (1996, 2nd ed., ) *8 ''Discovering Modern Set Theory. I: The Basics'', Winfried Just, Martin Weese (1996, ) *9 ''An Invitation to Arithmetic Geometry'', Dino Lorenzini (1996, ) *10 ''Representations of Finite and Compact Groups'', Barry Simon (199 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steinberg Group (K-theory)
In algebraic K-theory, a field of mathematics, the Steinberg group \operatorname(A) of a ring A is the universal central extension of the commutator subgroup of the stable general linear group of A . It is named after Robert Steinberg, and it is connected with lower K -groups, notably K_ and K_ . Definition Abstractly, given a ring A , the Steinberg group \operatorname(A) is the universal central extension of the commutator subgroup of the stable general linear group (the commutator subgroup is perfect and so has a universal central extension). Presentation using generators and relations A concrete presentation using generators and relations is as follows. Elementary matrices — i.e. matrices of the form (\lambda) := \mathbf + (\lambda) , where \mathbf is the identity matrix, (\lambda) is the matrix with \lambda in the (p,q) -entry and zeros elsewhere, and p \neq q — satisfy the following relations, called the Steinberg relations: : \begin e_(\lamb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm Residue Symbol
In number theory, a symbol is any of many different generalizations of the Legendre symbol. This article describes the relations between these various generalizations. The symbols below are arranged roughly in order of the date they were introduced, which is usually (but not always) in order of increasing generality. * Legendre symbol \left(\frac\right) defined for ''p'' a prime, ''a'' an integer, and takes values 0, 1, or −1. * Jacobi symbol \left(\frac\right) defined for ''b'' a positive odd integer, ''a'' an integer, and takes values 0, 1, or −1. An extension of the Legendre symbol to more general values of ''b''. * Kronecker symbol \left(\frac\right) defined for ''b'' any integer, ''a'' an integer, and takes values 0, 1, or −1. An extension of the Jacobi and Legendre symbols to more general values of ''b''. * Power residue symbol \left(\frac\right)=\left(\frac\right)_m is defined for ''a'' in some global field containing the ''m''th roots of 1 ( for some ''m''), ''b'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisible Group
In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an ''n''th multiple for each positive integer ''n''. Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups. Definition An abelian group (G, +) is divisible if, for every positive integer n and every g \in G, there exists y \in G such that ny=g. An equivalent condition is: for any positive integer n, nG=G, since the existence of y for every n and g implies that n G\supseteq G, and the other direction n G\subseteq G is true for every group. A third equivalent condition is that an abelian group G is divisible if and only if G is an injective object in the category of abelian groups; for this reason, a divisible group is sometimes called an injective group. An abelian group is p-divisible for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a ''generator'' of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology. Sometimes, real numbers R, and the complex numbers C (with their standard topologies) are also defined to be local fields; this is the convention we will adopt below. Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields. While Archimedean local fields have been quite well known in mathematics for at lea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold. Equivalent definitions By definition, a subset A of a topological space (X, \tau) is called if its complement X \setminus A is an open subset of (X, \tau); that is, if X \setminus A \in \tau. A set is closed in X if and only if it is equal to its closure in X. Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset A \subseteq X is always contained in its (topological) closure in X, which is denoted by \operatorname_X A; that is, if A \subseteq X then A \subseteq \oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the ''K''-groups of the integers. ''K''-theory was discovered in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only ''K''0, the zeroth ''K''-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of (higher) algebraic ''K''-theory through its links with motivic cohomology and specifically Chow groups. The subject also includes classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
