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Hasse–Arf Theorem
In mathematics, specifically in local class field theory, the Hasse–Arf theorem is a result concerning jumps of the upper numbering filtration of the Galois group of a finite Galois extension. A special case of it when the residue fields are finite was originally proved by Helmut Hasse, and the general result was proved by Cahit Arf. Statement Higher ramification groups The theorem deals with the upper numbered higher ramification groups of a finite abelian extension ''L''/''K''. So assume ''L''/''K'' is a finite Galois extension, and that ''v''''K'' is a discrete normalised valuation of ''K'', whose residue field has characteristic ''p'' > 0, and which admits a unique extension to ''L'', say ''w''. Denote by ''v''''L'' the associated normalised valuation ''ew'' of ''L'' and let \scriptstyle be the valuation ring of ''L'' under ''v''''L''. Let ''L''/''K'' have Galois group ''G'' and define the ''s''-th ramification group of ''L''/''K'' for any real ''s''&nb ... [...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] |
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 '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Numbering
In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension. Ramification theory of valuations In mathematics, the ramification theory of valuations studies the set of extensions of a valuation ''v'' of a field ''K'' to an extension ''L'' of ''K''. It is a generalization of the ramification theory of Dedekind domains. The structure of the set of extensions is known better when ''L''/''K'' is Galois. Decomposition group and inertia group Let (''K'', ''v'') be a valued field and let ''L'' be a finite Galois extension of ''K''. Let ''Sv'' be the set of equivalence classes of extensions of ''v'' to ''L'' and let ''G'' be the Galois group of ''L'' over ''K''. Then ''G'' acts on ''Sv'' by σ 'w''nbsp;= 'w'' ∘ σ(i.e. ''w'' is a representative of the equivalence class 'w''nbsp;∈ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
