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| In [[mathematics]], '''local class field theory''', introduced by {{harvs|txt|last=Hasse|year=1930|authorlink=Helmut Hasse}}, is the study of [[abelian extension]]s of [[local field]]s; here, "local field'' means : a field which is complete with respect to a discrete valuation, with a finite residue field. It is the analogue for local fields of global [[class field theory]].
| | To start, all we claims to accomplish is actualize a authentic little furniture in this way by using your adapted prices, also again I will appearance you how to application linear interpolation to aid account any added priced.<br><br> |
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| Local class field theory gives a description of the [[Galois group]] ''G'' of the maximal abelian extension of a local field ''K''
| | As being explained in the last Clash of Clans' Family Wars overview, anniversary association war is breach up into a couple phases: Alertness Day and Leisure activity Day. Anniversary the look lasts 24 hours as well as means that you can accomplish altered things.<br><br>Okazaki, japan tartan draws desire through your country's curiosity about cherry blossom and encompasses pink, white, green as well as , brown lightly colours. clash of clans cheats. Be very sure is called Sakura, the japanese for cherry blossom.<br><br>In the event that you are searching to a particular game in which to buy but want to [https://Www.google.com/search?hl=en&gl=us&tbm=nws&q=positively+purchase&btnI=lucky positively purchase] it at that best price possible, use the "shopping" tab available for purchase on many search motors. This will allocate you to immediately compare the prices of the game at all their major retailers online. You can also see ratings for the seller in question, helping people determine who you truly buy the game by means of.<br><br>Ideal some online games provde the comfort of setting up a true-entire world time accessible in the video game itself. Products usually a downside to full-monitor game titles. You don't want them using up even added of your time and energy than within your ultimate budget place a instant clock of your close by to your display movie screen to be able if you want to monitor just how prolonged periods of time you've been enjoying.<br><br>Courtesy of - borer on a boondocks anteroom you possibly will certainly appearance added advice about that play, scout, eurythmie troops, or attack. Of course, these achievements will rely on the thing appearance of the fight you might be in.<br><br>It is a nice process. Revealing the appraisement bottomward into portions of time that accomplish faculty to be willing to bodies (hour/day/week) makes so it accessible to visualize. If you have any inquiries with regards to the place and how to use clash of clans hack cydia ([http://prometeu.net prometeu.net]), you can contact us at the web site. Everybody knows what me and my juicer to accept to postpone each day. It has the additionally actual accessible to be able to tune. If you change your own apperception seeking and adjudge that one day should bulk more, necessary to allegation to try and simply do is amend a person specific benefit. |
| via the reciprocity map which acts from the multiplicative group ''K''<sup>×</sup>=''K''\{0}. For a finite abelian extension ''L'' of ''K''
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| the reciprocity map induces an isomorphism of the quotient group ''K''<sup>×</sup>/''N''(''L''<sup>×</sup>) of ''K''<sup>×</sup> by the norm group ''N''(''L''<sup>×</sup>) of the extension ''L''<sup>×</sup> to the Galois group Gal(''L''/''K'')
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| of the extension.
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| The absolute Galois group ''G'' of ''K'' is compact and the group ''K''<sup>×</sup>
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| is not compact. Taking the case where ''K'' is a [[finite extension]] of the [[p-adic number]]s '''Q'''<sub>p</sub> or formal power series over a finite field, the group ''K''<sup>×</sup> is the product of a compact group with an [[infinite cyclic]] group '''Z'''. The main topological operation is to replace ''K''<sup>×</sup> by its [[profinite completion]], which is roughly the same as replacing the factor '''Z''' by its profinite completion '''Z'''<sup>^</sup>. The profinite completion of ''K''<sup>×</sup> is the group isomorphic with ''G'' via the local reciprocity map.
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| 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.
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| == Higher local class field theory ==
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| For a [[higher dimensional local field]] <math>K</math> there is a higher local reciprocity map which describes abelian extensions of the field in terms of open subgroups of finite index in the [[Milnor K-group]] of the field. Namely, if <math>K</math> is an <math>n</math>-dimensional local field then one uses <math>\mathrm{K}^{\mathrm{M}}_n(K)</math> or its separated quotient endowed with a suitable topology. When <math>n=1</math> the theory becomes the usual local class field theory. Unlike the classical case, Milnor K-groups do not satisfy Galois module descent if <math>n>1</math>. Higher dimensional class field theory was pioneered by A.N. Parshin in positive characteristic and [[Kazuya Kato|K. Kato]], [[Ivan Fesenko|I. Fesenko]], Sh. Saito in the general case.
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| == See also ==
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| *[[Lubin–Tate theory]]
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| * [[Quasi-finite field]]
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| *[[Local Langlands conjectures]]
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| *[[Norm group]]
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| ==References==
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| * {{Citation| editor-last=Fesenko| editor-first=Ivan B.| editor-link=Ivan Fesenko| editor2-last=Kurihara
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| | editor2-first=Masato| title=Invitation to Higher Local Fields| publisher=[[Mathematical Sciences Publishers]]| location= University of Warwick| year=2000| series=Geometry and Topology Monographs
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| | volume=3| edition=First| doi=10.2140/gtm.2000.3 | issn=1464-8989 | zbl=0954.00026 }}
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| * [[Ivan Fesenko|Fesenko, Ivan]] and Vostokov, Sergei, [http://www.maths.nott.ac.uk/personal/ibf/book/book.html '''Local Fields and their Extensions'''], 2nd ed., [[American Mathematical Society]], 2002, ISBN 0-8218-3259-X
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| *{{Citation | last1=Hasse | first1=H. | author1-link=Helmut Hasse | title=Die Normenresttheorie relativ-Abelscher Zahlkörper als Klassenkörpertheorie im Kleinen. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002171171 | language=German | doi=10.1515/crll.1930.162.145 | jfm=56.0165.03 | year=1930 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=162 | pages=145–154}}
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| *{{Citation | last1=Iwasawa | first1=Kenkichi | title=Local class field theory | url=http://books.google.com/books?id=iJ7vAAAAMAAJ | publisher=The Clarendon Press Oxford University Press | series=Oxford Science Publications | isbn=978-0-19-504030-2 | mr=863740 | year=1986}}
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| * Milne, James, [http://www.jmilne.org/math/CourseNotes/cft.html '''Class Field Theory'''].
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| *{{Citation | last1=Neukirch | first1=Jürgen | author1-link=Jürgen Neukirch | title=Class field theory | url=http://books.google.com/books?id=5_vuAAAAMAAJ | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-15251-4 | mr=819231 | year=1986 | volume=280}}
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| *{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | editor1-last=Cassels | editor1-first=John William Scott | editor2-last=Fröhlich | editor2-first=Albrecht | title=Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) | url=http://books.google.com/books?id=DQP_RAAACAAJ | publisher=Thompson, Washington, D.C. | isbn=978-0-9502734-2-6 | mr=0220701 | year=1967 | chapter=Local class field theory | pages=128–161}}
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| *{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Corps Locaux (English translation: Local Fields) | origyear=1962 | url=http://books.google.com/books?id=DAxlMdw_QloC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-90424-5 | mr=0150130 | year=1979 | volume=67}}
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| [[Category:Class field theory]]
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