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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 a finite extension of 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'' ... [...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 points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Sciences Publishers
Mathematical Sciences Publishers is a nonprofit publishing company run by and for mathematicians. It publishes several journals and the book series ''Geometry & Topology Monographs''. It is run from a central office in the Department of Mathematics at the University of California, Berkeley. Journals owned and published * ''Algebra & Number Theory'' * ''Algebraic & Geometric Topology'' * ''Analysis & PDE'' * ''Annals of K-Theory'' * ''Communications in Applied Mathematics and Computational Science'' * ''Geometry & Topology'' * ''Innovations in Incidence Geometry—Algebraic, Topological and Combinatorial'' * ''Involve, a Journal of Mathematics'' * ''Journal of Algebraic Statistics'' * ''Journal of Mechanics of Materials and Structures'' * ''Journal of Software for Algebra and Geometry'' * ''Mathematics and Mechanics of Complex Systems'' * ''Moscow Journal of Combinatorics and Number Theory'' * ''Pacific Journal of Mathematics The Pacific Journal of Mathematics is a mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm Group
In number theory, a norm group is a group of the form N_(L^\times) where L/K is a finite abelian extension of nonarchimedean local fields. One of the main theorems in local class field theory states that the norm groups in K^\times are precisely the open subgroups of K^\times of finite index. See also *Takagi existence theorem {{short description, Correspondence between finite abelian extensions and generalized ideal class groups In class field theory, the Takagi existence theorem states that for any number field ''K'' there is a one-to-one inclusion reversing correspond ... References *J.S. Milne, ''Class field theory.'' Version 4.01. Algebraic number theory {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Langlands Conjectures
In mathematics, the local Langlands conjectures, introduced by , are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group ''G'' over a local field ''F'', and representations of the Langlands group of ''F'' into the L-group of ''G''. This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non-abelian Galois groups. Local Langlands conjectures for GL1 The local Langlands conjectures for GL1(''K'') follow from (and are essentially equivalent to) local class field theory. More precisely the Artin map gives an isomorphism from the group GL1(''K'')= ''K''* to the abelianization of the Weil group. In particular irreducible smooth representations of GL1(''K'') are 1-dimensional as the group is abelian, so can be identified with homomorphisms of the Weil group to GL1(C). This gives the Langlands corresponden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher Class Field Theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local field, local and global field, global fields using objects associated to the ground field. David Hilbert, Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Leopold Kronecker, Kronecker and it was actually Eduard Ritter von Weber, Weber who coined the term before Hilbert's fundamental papers came out. The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Teiji Takagi, Takagi and Emil Artin, Artin (with the help of Chebotarev's theorem). One of the major results is: given a number field ''F'', and writing ''K'' for the Hilbert class field, maximal abelian unramified extension of ''F'', the Galois group of ''K'' over ''F'' is canonically isomorphic to the ideal clas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kazuya Kato
is a Japanese mathematician. He grew up in the prefecture of Wakayama in Japan. He attended college at the University of Tokyo, from which he also obtained his master's degree in 1975, and his PhD in 1980. He was a professor at Tokyo University, Tokyo Institute of Technology and Kyoto University. He joined the faculty of the University of Chicago in 2009. He has contributed to number theory and related parts of algebraic geometry. His first work was in the higher-dimensional generalisations of local class field theory using algebraic K-theory. His theory was then extended to higher global class field theory in which several of his papers were written jointly with Shuji Saito. He contributed to various other areas such as ''p''-adic Hodge theory, logarithmic geometry (he was one of its creators together with Jean-Marc Fontaine and Luc Illusie), comparison conjectures, special values of zeta functions including applications to the Birch-Swinnerton-Dyer conjecture, the B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor K-group
In mathematics, Milnor K-theory is an algebraic invariant (denoted K_*(F) for a field F) defined by as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebraic and give some insight about its relationships with other parts of mathematics, such as Galois cohomology and the Grothendieck–Witt ring of quadratic forms. Before Milnor K-theory was defined, there existed ad-hoc definitions for K_1 and K_2. Fortunately, it can be shown Milnor is a part of algebraic , which in general is the easiest part to compute. Definition Motivation After the definition of the Grothendieck group K(R) of a commutative ring, it was expected there should be an infinite set of invariants K_i(R) called higher groups, from the fact there exists a short exact sequence :K(R,I) \to K(R) \to K(R/I) \to 0 which should have a continuation by a long exact sequence. Note the group on the left is relative . This led to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 many ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michiel Hazewinkel
Michiel Hazewinkel (born 22 June 1943) is a Dutch mathematician, and Emeritus Professor of Mathematics at the Centre for Mathematics and Computer Science and the University of Amsterdam, particularly known for his 1978 book ''Formal groups and applications'' and as editor of the ''Encyclopedia of Mathematics''. Biography Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam. He received his BA in mathematics and physics in 1963, his MA in mathematics with a minor in philosophy in 1965 and his PhD in 1969 under supervision of Frans Oort and Albert Menalda for the thesis "Maximal Abelian Extensions of Local Fields".Michiel Hazewinkel, Curriculum vitae at michhaz.home.xs4all.nl. Accessed September 10, 2013 After graduation Hazewinkel started hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jürgen Neukirch
Jürgen Neukirch (24 July 1937 – 5 February 1997) was a German mathematician known for his work on algebraic number theory. Education and career Neukirch received his diploma in mathematics in 1964 from the University of Bonn. For his Ph.D. thesis, written under the direction of Wolfgang Krull, he was awarded in 1965 the Felix-Hausdorff-Gedächtnis-Preis. He completed his habilitation one year later. From 1967 to 1969 he was guest professor at Queen's University in Kingston, Ontario and at the Massachusetts Institute of Technology in Cambridge, Massachusetts, after which he was a professor in Bonn. In 1971 he became a professor at the University of Regensburg. Contributions He is known for his work on the embedding problem in algebraic number theory, the Báyer–Neukirch theorem on special values of L-functions, arithmetic Riemann existence theorems and the Neukirch–Uchida theorem in birational anabelian geometry. He gave a simple description of the reciprocity maps in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
