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Higher Local Field
In mathematics, a higher (-dimensional) local field is an important example of a complete discrete valuation field. Such fields are also sometimes called multi-dimensional local fields. On the usual local fields (typically completions of number fields or the quotient fields of local rings of algebraic curves) there is a unique surjective discrete valuation (of rank 1) associated to a choice of a local parameter of the fields, unless they are archimedean local fields such as the real numbers and complex numbers. Similarly, there is a discrete valuation of rank ''n'' on almost all ''n''-dimensional local fields, associated to a choice of ''n'' local parameters of the field. In contrast to one-dimensional local fields, higher local fields have a sequence of residue fields.Fesenko, I., Kurihara, M. (eds.) ''Invitation to Higher Local Fields''. Geometry and Topology Monographs, 2000, section 1 (Zhukov). There are different integral structures on higher local fields, depending how man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Valuation
In mathematics, a discrete valuation is an integer valuation on a field ''K''; that is, a function: :\nu:K\to\mathbb Z\cup\ satisfying the conditions: :\nu(x\cdot y)=\nu(x)+\nu(y) :\nu(x+y)\geq\min\big\ :\nu(x)=\infty\iff x=0 for all x,y\in K. Note that often the trivial valuation which takes on only the values 0,\infty is explicitly excluded. A field with a non-trivial discrete valuation is called a discrete valuation field. Discrete valuation rings and valuations on fields To every field K with discrete valuation \nu we can associate the subring ::\mathcal_K := \left\ of K, which is a discrete valuation ring. Conversely, the valuation \nu: A \rightarrow \Z\cup\ on a discrete valuation ring A can be extended in a unique way to a discrete valuation on the quotient field K=\text(A); the associated discrete valuation ring \mathcal_K is just A. Examples * For a fixed prime p and for any element x \in \mathbb different from zero write x = p^j\frac with j, a,b \in \Z such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flag (geometry)
In (polyhedral) geometry, a flag is a sequence of Face (geometry), faces of a Abstract polytope, polytope, each contained in the next, with exactly one face from each dimension. More formally, a flag of an -polytope is a set such that and there is precisely one in for each , Since, however, the minimal face and the maximal face must be in every flag, they are often omitted from the list of faces, as a shorthand. These latter two are called improper faces. For example, a flag of a polyhedron comprises one Vertex (geometry), vertex, one Edge (geometry), edge incident to that vertex, and one polygonal face incident to both, plus the two improper faces. A polytope is regular polytope, regular if, and only if, its symmetry group is transitive group action, transitive on its flags. This definition excludes chiral polytopes. Two flags are -adjacent if they only differ by a face of rank . They are adjacent if they are -adjacent for some value of . Each flag is -adjacent to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian group, abelian. When the Galois group is also cyclic group, cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable group, solvable, i.e., if the group can be decomposed into a series of normal Group extension, extensions of an abelian group. Every finite extension of a finite field is a cyclic extension. Description Class field theory provides detailed information about the abelian extensions of number fields, function field of an algebraic variety, function fields of algebraic curves over finite fields, and local fields. There are two slightly different definitions of the term cyclotomic extension. It can mean either an extension formed by adjoining roots of unity to a field, or a subextension of such an extension. The cyclotomic fields are examples. A cyclotomic extension, under either d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory. Definition Suppose that E is an extension of the field F (written as E/F and read "''E'' over ''F''). An automorphism of E/F is defined to be an automorphism of E that fixes F pointwise. In other words, an automorphism of E/F is an isomorphism \alpha:E\to E such that \alpha(x) = x for each x\in F. The set of all automorphisms of E/F forms a group with the operation of function composition. This group is sometimes denoted by \operatorname(E/F). If E/F is a Galois extension, then \op ... [...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 classi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Class Field Theory
In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the ''p''-adic numbers Q''p'' (where ''p'' is any prime number), or the field of formal Laurent series F''q''((''T'')) over a finite field F''q''. Approaches to local class field theory Local class field theory gives a description of the Galois group ''G'' of the maximal abelian extension of a local field ''K'' via the reciprocity map which acts from the multiplicative group ''K''×=''K''\. For a finite abelian extension ''L'' of ''K'' the reciprocity map induces an isomorphism of the quotient group ''K''×/''N''(''L''×) of ''K''× by the norm group ''N''(''L''×) of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Analysis
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis. Harmonic analysis has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis, spectral analysis, and neuroscience. The term "harmonics" originated from the Ancient Greek word ''harmonikos'', meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the freq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haar Measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of mathematical analysis, analysis, number theory, group theory, representation theory, mathematical statistics, statistics, probability theory, and ergodic theory. Preliminaries Let (G, \cdot) be a locally compact space, locally compact Hausdorff space, Hausdorff topological group. The Sigma-algebra, \sigma-algebra generated by all open subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g is an element of G and S is a subset of G, then we define the left and right Coset, translates of S by ''g'' as follows: * Left ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Additive Measure
In mathematics, in particular in measure theory, a content \mu is a real-valued function defined on a collection of subsets \mathcal such that # \mu(A)\in\ , \infty\text A \in \mathcal. # \mu(\varnothing) = 0. # \mu\Bigl(\bigcup_^n A_i\Bigr) = \sum_^n \mu(A_i) \text A_1, \dots, A_n, \bigcup_^n A_i \in \mathcal \text A_i \cap A_j = \varnothing \text i \neq j. That is, a content is a generalization of a measure: while the latter must be countably additive, the former must only be finitely additive. In many important applications the \mathcal is chosen to be a ring of sets or to be at least a semiring of sets in which case some additional properties can be deduced which are described below. For this reason some authors prefer to define contents only for the case of semirings or even rings. If a content is additionally ''σ''-additive it is called a pre-measure and if furthermore \mathcal is a ''σ''-algebra, the content is called a measure. Therefore, every (real-valued) measure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Measure
In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, and a difference of slopes is invariant under shear mapping. Ergodic theory is the study of invariant measures in dynamical systems. The Krylov–Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration. Definition Let (X, \Sigma) be a measurable space and let f : X \to X be a measurable function from X to itself. A measure \mu on (X, \Sigma) is said to be invariant under f if, for every measurable set A in \Sigma, \mu\left(f^(A)\right) = \mu(A). In terms of the pushforward measure, this states that f_*(\mu) = \mu. The collection of measures (usually probability measures) on X that are invariant under f is sometimes denoted M_f(X). The collection of ergod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ind-completion
In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category ''C''. The objects in this ind-completed category, denoted Ind(''C''), are known as direct systems, they are functors from a small filtered category ''I'' to ''C''. The dual concept is the pro-completion, Pro(''C''). Definitions Filtered categories Direct systems depend on the notion of ''filtered categories''. For example, the category N, whose objects are natural numbers, and with exactly one morphism from ''n'' to ''m'' whenever n \le m, is a filtered category. Direct systems A ''direct system'' or an ''ind-object'' in a category ''C'' is defined to be a functor :F : I \to C from a small filtered category ''I'' to ''C''. For example, if ''I'' is the category N mentioned above, this datum is equivalent to a sequence :X_0 \to X_1 \to \cdots of objects in ''C'' together with morphisms as displayed. The ind-completion Ind-objects in ''C'' form a cat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
