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Contou-Carrère Symbol
In mathematics, the Contou-Carrère symbol 〈''a'',''b''〉 is a Steinberg symbol defined on pairs of invertible elements of the ring of Laurent power series over an Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are ... ''k'', taking values in the group of units of ''k''. It was introduced by . Definition If ''k'' is an Artinian local ring, then any invertible formal Laurent series ''a'' with coefficients in ''k'' can be written uniquely as :a=a_0t^\prod_(1-a_it^i) where ''w''(''a'') is an integer, the elements ''a''''i'' are in ''k'', and are in ''m'' if ''i'' is negative, and is a unit if ''i'' = 0. The Contou-Carrère symbol 〈''a'',''b''〉 of ''a'' and ''b'' is defined to be :\langle a,b\rangle=(-1)^\frac References * Number theory {{numthe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Hideya 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 symbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit (ring Theory)
In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this property and is called the multiplicative inverse of . The set of units of forms a group under multiplication, called the group of units or unit group of . Other notations for the unit group are , , and (from the German term ). Less commonly, the term ''unit'' is sometimes used to refer to the element of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also unit matrix. Because of this ambiguity, is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng. Examples The multiplicative identity and its additive inverse are always units. More ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent Series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass had previously described it in a paper written in 1841 but not published until 1894. Definition The Laurent series for a complex function f(z) about an arbitrary point c is given by f(z) = \sum_^\infty a_n(z-c)^n, where the coefficients a_n are defined by a contour integral that generalizes Cauchy's integral formula: a_n =\frac\oint_\gamma \frac \, dz. The path of integration \gamma is counterclockwise around a Jordan curve enclosing c and lying in an annulus A in which f(z) is holomorphic ( analytic). The expansion for f(z) will then be valid anywhere inside the annulus. The annulus is shown in red in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artinian Ring
In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition. Precisely, a ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian. For commutative rings the left and right definitions coincide, but in general they are distinct from each other. The Wedderburn–Artin theorem ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
