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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''. It is the analogue for local fields of global class field theory. ==Connection to Galois groups== 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 the extension ''L''× to the Galois group Gal(''L''/''K'') of the extension.〔Fesenko, Ivan and Vostokov, Sergei, (Local Fields and their Extensions ), 2nd ed., American Mathematical Society, 2002, ISBN 0-8218-3259-X〕 The absolute Galois group ''G'' of ''K'' is compact and the group ''K''× is not compact. Taking the case where ''K'' is a finite extension of the p-adic numbers Qp or formal power series over a finite field, the group ''K''× is the product of a compact group with an infinite cyclic group Z. The main topological operation is to replace ''K''× by its profinite completion, which is roughly the same as replacing the factor Z by its profinite completion Z^. The profinite completion of ''K''× is the group isomorphic with ''G'' via the local reciprocity map. The actual isomorphism used and the existence theorem is described in the theory of the norm residue symbol. There are several different approaches to the theory, using central division algebras or Tate cohomology or an explicit description of the reciprocity map. There are also two different normalizations of the reciprocity map: in the case of an unramified extension, one of them asks that the (arithmetic) Frobenius element corresponds to the elements of "K" of valuation 1; the other one is the opposite. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「local class field theory」の詳細全文を読む スポンサード リンク
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