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| In [[mathematics]], in the field of [[algebraic number theory]], a '''modulus''' (plural '''moduli''') (or '''cycle''',<ref>{{harvnb|Lang|1994|loc=§VI.1}}</ref> or '''extended ideal'''<ref>{{harvnb|Cohn|1985|loc=definition 7.2.1}}</ref>) is a formal product of [[Place (mathematics)|place]]s of a [[global field]] (i.e. an [[algebraic number field]] or a [[global function field]]). It is used to encode [[ramification]] data for [[abelian extension]]s of a global field. See [[Modulo operation]] for a definition most people will be seeking.
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| ==Definition==
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| Let ''K'' be a global field with ring of integers ''R''. A '''modulus''' is a formal product<ref>{{harvnb|Janusz|1996|loc=§IV.1}}</ref><ref>{{harvnb|Serre|1988|loc=§III.1}}</ref> | |
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| :<math>\mathbf{m} = \prod_{\mathbf{p}} \mathbf{p}^{\nu(\mathbf{p})},\,\,\nu(\mathbf{p})\geq0 </math>
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| where '''p''' runs over all [[place (mathematics)|places]] of ''K'', [[finite place|finite]] or [[infinite place|infinite]], the exponents ν('''p''') are zero except for finitely many '''p'''. If ''K'' is a number field, ν('''p''') = 0 or 1 for real places and ν('''p''') = 0 for complex places. If ''K'' is a function field, ν('''p''') = 0 for all infinite places.
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| In the function field case, a modulus is the same thing as an [[effective divisor]],<ref>{{harvnb|Serre|1988|loc=§III.1}}</ref> and in the number field case, a modulus can be considered as special form of [[Arakelov divisor]].<ref>{{harvnb|Neukirch|1999|loc=§III.1}}</ref>
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| The notion of [[modular arithmetic|congruence]] can be extended to the setting of moduli. If ''a'' and ''b'' are elements of ''K''<sup>×</sup>, the definition of ''a'' ≡<sup>∗</sup>''b'' (mod '''p'''<sup>ν</sup>) depends on what type of prime '''p''' is:<ref>{{harvnb|Janusz|1996|loc=§IV.1}}</ref><ref>{{harvnb|Serre|1988|loc=§III.1}}</ref> | |
| *if it is finite, then
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| ::<math>a\equiv^\ast\!b\,(\mathrm{mod}\,\mathbf{p}^\nu)\Leftrightarrow \mathrm{ord}_\mathbf{p}\left(\frac{a}{b}-1\right)\geq\nu</math>
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| :where ord<sub>'''p'''</sub> is the [[normalized valuation]] associated to '''p''';
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| *if it is a real place (of a number field) and ν = 1, then
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| ::<math>a\equiv^\ast\!b\,(\mathrm{mod}\,\mathbf{p})\Leftrightarrow \frac{a}{b}>0</math>
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| :under the [[real embedding]] associated to '''p'''.
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| *if it is any other infinite place, there is no condition.
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| Then, given a modulus '''m''', ''a'' ≡<sup>∗</sup>''b'' (mod '''m''') if ''a'' ≡<sup>∗</sup>''b'' (mod '''p'''<sup>ν('''p''')</sup>) for all '''p''' such that ν('''p''') > 0.
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| ==Ray class group==
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| The '''ray modulo m''' is<ref>{{harvnb|Milne|2008|loc=§V.1}}</ref><ref>{{harvnb|Janusz|1996|loc=§IV.1}}</ref><ref>{{harvnb|Serre|1988|loc=§VI.6}}</ref>
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| :<math>K_{\mathbf{m},1}=\left\{ a\in K^\times : a\equiv^\ast\!1\,(\mathrm{mod}\,\mathbf{m})\right\}.</math> | |
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| A modulus '''m''' can be split into two parts, '''m'''<sub>f</sub> and '''m'''<sub>∞</sub>, the product over the finite and infinite places, respectively. Let ''I''<sup>'''m'''</sup> to be one of the following:
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| *if ''K'' is a number field, the subgroup of the [[group of fractional ideals]] generated by ideals coprime to '''m'''<sub>f</sub>;<ref>{{harvnb|Janusz|1996|loc=§IV.1}}</ref>
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| *if ''K'' is a function field of an [[algebraic curve]] over ''k'', the group of divisors, [[rational divisor|rational]] over ''k'', with [[support of a divisor|support]] away from '''m'''.<ref>{{harvnb|Serre|1988|loc=§V.1}}</ref>
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| In both case, there is a [[group homomorphism]] ''i'' : ''K''<sub>'''m''',1</sub> → ''I''<sup>'''m'''</sup> obtained by sending ''a'' to the [[principal ideal]] (resp. [[principal divisor|divisor]]) (''a'').
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| The '''ray class group modulo m''' is the quotient ''C''<sub>'''m'''</sub> = ''I''<sup>'''m'''</sup> / i(''K''<sub>'''m''',1</sub>).<ref>{{harvnb|Janusz|1996|loc=§IV.1}}</ref><ref>{{harvnb|Serre|1988|loc=§VI.6}}</ref> A coset of i(''K''<sub>'''m''',1</sub>) is called a '''ray class modulo m'''.
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| [[Erich Hecke]]'s original definition of [[Hecke character]]s may be interpreted in terms of [[Character (mathematics)|character]]s of the ray class group with respect to some modulus '''m'''.<ref>{{harvnb|Neukirch|1999|loc=§VII.6}}</ref>
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| ===Properties===
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| When ''K'' is a number field, the following properties hold.<ref>{{harvnb|Janusz|1996|§4.1}}</ref>
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| * When '''m''' = 1, the ray class group is just the [[ideal class group]].
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| * The ray class group is finite. Its order is the '''ray class number'''.
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| * The ray class number is divisible by the [[Class number (number theory)|class number]] of ''K''.
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| *{{Citation
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| | last=Cohn
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| | first=Harvey
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| | title=Introduction to the construction of class fields
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| | series=Cambridge studies in advanced mathematics
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| | volume=6
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| | publisher=[[Cambridge University Press]]
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| | year=1985
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| | isbn=978-0-521-24762-7
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| }}
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| *{{Citation
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| | last=Janusz
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| | first=Gerald J.
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| | title=Algebraic number fields
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| | publisher=[[American Mathematical Society]]
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| | series=Graduate Studies in Mathematics
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| | volume=7
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| | year=1996
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| | isbn=978-0-8218-0429-2
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| }}
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| *{{Citation
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| | last=Lang
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| | first=Serge
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| | author-link=Serge Lang
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| | title=Algebraic number theory
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| | edition=2
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| | publisher=[[Springer-Verlag]]
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| | year=1994
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| | series=[[Graduate Texts in Mathematics]]
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| | volume=110
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| | place=New York
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| | isbn=978-0-387-94225-4
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| | mr=1282723
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| }}
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| *{{Citation
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| | last=Milne
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| | first=James
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| | title=Class field theory
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| | url=http://jmilne.org/math/CourseNotes/cft.html
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| | edition=v4.0
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| | year=2008
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| | accessdate=2010-02-22
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| }}
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| *{{Neukirch ANT}}
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| *{{Citation
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| | last=Serre
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| | first=Jean-Pierre
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| | author-link=Jean-Pierre Serre
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| | title=Algebraic groups and class fields
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| | year=1988
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| | isbn=978-0-387-96648-9
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| | publisher=[[Springer-Verlag]]
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| | location=New York
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| | series=[[Graduate Texts in Mathematics]]
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| | volume=117
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| }}
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| {{DEFAULTSORT:Modulus (Algebraic Number Theory)}}
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| [[Category:Algebraic number theory]]
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Let me first you must do introducing my own self. My name is Sidney. His day job is often a production and planning officer but soon he'll be on her own. The favorite hobby for her and her kids is bottle tops collecting but she doesn't contain the time most recently. Rhode Island is is a good idea place he's been basically and quality guy never cross. You can always find her website here: http://www.quora.com/what is financial spread betting-is-financial-spread-betting