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| In [[mathematics]], a '''quasi-finite field'''<ref>{{harv|Artin|Tate|2009|loc=§XI.3}} say that the field satisfies "Moriya's axiom"</ref> is a generalisation of a [[finite field]]. Standard [[local class field theory]] usually deals with [[complete valued field]]s whose residue field is ''finite'' (i.e. [[non-archimedean local field]]s), but the theory applies equally well when the residue field is only assumed quasi-finite.<ref>As shown by Mikao Moriya {{harv|Serre|1979|loc=chapter XIII, p. 188}}</ref>
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| == Formal definition ==
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| A '''quasi-finite field''' is a [[perfect field]] ''K'' together with an [[isomorphism]] of [[topological group]]s
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| : <math>\phi : \hat{\mathbf Z} \to \operatorname{Gal}(K_s/K),</math>
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| where ''K''<sub>''s''</sub> is an [[algebraic closure]] of ''K'' (necessarily separable because ''K'' is perfect). The [[field extension]] ''K''<sub>''s''</sub>/''K'' is infinite, and the [[Galois group]] is accordingly given the [[Krull topology]]. The group <math>\widehat{\mathbf{Z}}</math> is the [[profinite completion]] of [[integer]]s with respect to its subgroups of finite index.
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| This definition is equivalent to saying that ''K'' has a unique (necessarily [[cyclic extension|cyclic]]) extension ''K''<sub>''n''</sub> of degree ''n'' for each integer ''n'' ≥ 1, and that the union of these extensions is equal to ''K''<sub>''s''</sub>.<ref>{{harv|Serre|1979|loc=§XIII.2 exercise 1, p. 192}}</ref> Moreover, as part of the structure of the quasi-finite field, there is a generator ''F''<sub>''n''</sub> for each Gal(''K''<sub>''n''</sub>/''K''), and the generators must be ''coherent'', in the sense that if ''n'' divides ''m'', the restriction of ''F''<sub>''m''</sub> to ''K''<sub>''n''</sub> is equal to ''F''<sub>''n''</sub>.
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| == Examples ==
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| The most basic example, which motivates the definition, is the finite field ''K'' = '''GF'''(''q''). It has a unique cyclic extension of degree ''n'', namely ''K''<sub>''n''</sub> = '''GF'''(''q''<sup>''n''</sup>). The union of the ''K''<sub>''n''</sub> is the algebraic closure ''K''<sub>''s''</sub>. We take ''F''<sub>''n''</sub> to be the [[Frobenius element]]; that is, ''F''<sub>''n''</sub>(''x'') = ''x''<sup>''q''</sup>.
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| Another example is ''K'' = '''C'''((''T'')), the ring of [[formal Laurent series]] in ''T'' over the field '''C''' of [[complex number]]s. (These are simply [[formal power series]] in which we also allow finitely many terms of negative degree.) Then ''K'' has a unique cyclic extension
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| : <math>K_n = \mathbf C((T^{1/n}))</math>
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| of degree ''n'' for each ''n'' ≥ 1, whose union is an algebraic closure of ''K'' called the field of [[Puiseux series]], and that a generator of Gal(''K''<sub>''n''</sub>/''K'') is given by
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| : <math>F_n(T^{1/n}) = e^{2\pi i/n} T^{1/n}.</math>
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| This construction works if '''C''' is replaced by any algebraically closed field ''C'' of characteristic zero.<ref>{{harv|Serre|1979|loc=§XIII.2, p. 191}}</ref>
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * {{Citation
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| | last=Artin
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| | first=Emil
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| | author-link=Emil Artin
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| | last2=Tate
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| | first2=John
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| | author2-link=John Tate
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| | title=Class field theory
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| | publisher=[[American Mathematical Society]]
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| | year=2009
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| | origyear=1967
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| | isbn=978-0-8218-4426-7
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| | mr=2467155 | zbl=1179.11040
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| }}
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| * {{citation | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Local fields | others=Translated from the French by Marvin Jay Greenberg | series=[[Graduate Texts in Mathematics]] | volume=67 | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 | mr=554237 }}
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| [[Category:Class field theory]]
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| [[Category:Field theory]]
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