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| In [[algebraic number theory]], the '''different ideal''' (sometimes simply the '''different''') is defined to measure the (possible) lack of duality in the [[ring of integers]] of an [[algebraic number field]] ''K'', with respect to the [[field trace]]. It then encodes the [[Ramification#In_algebraic_number_theory|ramification]] data for [[prime ideal]]s of the ring of integers. It was introduced by [[Richard Dedekind]] in 1882.<ref>{{harvnb|Dedekind|1882}}</ref><ref>{{harvnb|Bourbaki|1994}}, p. 102</ref>
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| ==Definition==
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| If ''O''<sub>''K''</sup> is the ring of integers of ''K'', and ''tr'' denotes the field trace from ''K'' to the [[rational number field]] '''Q''', then
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| :<math> x \mapsto \operatorname{tr}~x^2 </math>
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| is an [[integral quadratic form]] on ''O''<sub>''K''</sup>. Its [[Discriminant of a quadratic form|discriminant]] as quadratic form need not be +1 (in fact this happens only for the case ''K'' = '''Q'''). Define the ''inverse different'' or ''codifferent''<ref name=Serre50>{{harvnb|Serre|1979|p=50}}</ref><ref name=FT125>{{harvnb|Fröhlich|Taylor|1991|p=125}}</ref> or ''Dedekind's complementary module''<ref name=Neukirch195>{{harvnb|Neukirch|1999|p=195}}</ref> as the set ''I'' of ''x'' ∈ ''K'' such that tr(''xy'') is an integer for all ''y'' in ''O''<sub>''K''</sup>, then ''I'' is a [[fractional ideal]] of ''K'' containing ''O''<sub>''K''</sup>. By definition, the '''different ideal''' δ<sub>''K''</sub> is the inverse fractional ideal ''I''<sup>−1</sup>: it is an ideal of ''O''<sub>''K''</sup>.
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| The [[ideal norm]] of ''δ''<sub>''K''</sub> is equal to the ideal of ''Z'' generated by the [[Discriminant of an algebraic number field|field discriminant]] ''D''<sub>''K''</sub> of ''K''.
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| The ''different of an element'' α of ''K'' with minimal polynomial ''f'' is defined to be δ(α) = ''f''′(α) if α generates the field ''K'' (and zero otherwise):<ref name=Nark160/> we may write
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| :<math>\delta(\alpha) = \prod \left({\alpha - \alpha^{(i)}}\right) \ </math> | |
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| where the α<sup>(''i'')</sup> run over all the roots of the characteristic polynomial of α other than α itself.<ref name=Hecke116>{{harvnb|Hecke|1981|p=116}}</ref> The different ideal is generated by the differents of all integers α in ''O''<sub>''K''</sup>.<ref name=Nark160>{{harvnb|Narkiewicz|1990|p=160}}</ref><ref name=Hecke121>{{harvnb|Hecke|1981|p=121}}</ref> This is Dedekind's original definition.<ref name=Neukirch1978>{{harvnb|Neukirch|1999|pp=197–198}}</ref>
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| The different is also defined for an [[Finite extensions of local fields|finite degree extension]] of [[local field]]s. It plays a basic role in [[Pontryagin duality]] for [[p-adic field]]s.
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| ==Relative different==
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| The '''relative different''' δ<sub>''L'' / ''K''</sub> is defined in a similar manner for an extension of number fields ''L'' / ''K''. The [[relative norm]] of the relative different is then equal to the relative discriminant Δ<sub>''L'' / ''K''</sub>.<ref name=Neukirch201>{{harvnb|Neukirch|1999|p=201}}</ref> In a [[tower of fields]] ''L'' / ''K'' / ''F'' the relative differents are related by δ<sub>''L'' / ''F''</sub> = δ<sub>''L'' / ''K''</sub>''δ''<sub>''K'' / ''F''</sub>.<ref name=Neukirch195/><ref name=FT126>{{harvnb|Fröhlich|Taylor|1991|p=126}}</ref>
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| The relative different equals the annihilator of the relative [[Kähler differential]] module <math>\Omega^1_{O_L/O_K}</math>:<ref name=Neukirch201/><ref name=Serre59>{{harvnb|Serre|1979|p=59}}</ref>
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| <math>\delta_{L/K} = \{ x \in O_L : x \mathrm{d} y = 0 \text{ for all } y \in O_L \} . </math>
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| The [[ideal class]] of the relative different δ<sub>''L'' / ''K''</sub> is always a square in the [[class group]] of ''O''<sub>''L''</sub>, the ring of integers of ''L''.<ref name=Hecke2346>{{harvnb|Hecke|1981|pp=234-236}}</ref> Since the relative discriminant is the norm of the relative different it is the square of a class in the class group of ''O''<sub>''K''</sub>:<ref name=Nark304>{{harvnb|Narkiewicz|1990|p=304}}</ref> indeed, it is the square of the [[Steinitz class]] for ''O''<sub>''L''</sub> as a ''O''<sub>''K''</sub>-module.<ref name=Nark401>{{harvnb|Narkiewicz|1990|p=401}}</ref>
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| ==Ramification==
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| The relative different encodes the [[ramification#In algebraic number theory|ramification]] data of the field extension ''L'' / ''K''. A prime ideal ''p'' of ''K'' ramifies in ''L'' if the factorisation of ''p'' in ''L'' contains a prime of ''L'' to a power higher than 1: this occurs if and only if ''p'' divides the relative discriminant Δ<sub>''L'' / ''K''</sub>. More precisely, if
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| :''p'' = ''P''<sub>1</sub><sup>''e''(1)</sup> ... ''P''<sub>''k''</sub><sup>''e''(''k'')</sup> | |
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| is the factorisation of ''p'' into prime ideals of ''L'' then ''P''<sub>''i''</sub> divides the relative different δ<sub>''L'' / ''K''</sub> if and only if ''P''<sub>''i''</sub> is ramified, that is, if and only if the ramification index ''e''(''i'') is greater than 1.<ref name=FT126/><ref name=Neukirch199>{{harvnb|Neukirch|1999|pp=199}}</ref> The precise exponent to which a ramified prime ''P'' divides δ is termed the '''differential exponent''' of '''P''' and is equal to ''e'' − 1 if ''P'' is [[tamely ramified]]: that is, when ''P'' does not divide ''e''.<ref name=Nark166>{{harvnb|Narkiewicz|1990|p=166}}</ref> In the case when ''P'' is [[wildly ramified]] the differential exponent lies in the range ''e'' to ''e'' + ν<sub>''P''</sub>(e) − 1.<ref name=Neukirch199/><ref>{{harvnb|Weiss|1976}}, p. 114</ref><ref name=Nark194270>{{harvnb|Narkiewicz|1990|pp=194,270}}</ref> The differential exponent can be computed from the orders of the [[Hasse–Arf theorem#Higher ramification groups|higher ramification groups]] for Galois extensions:<ref>{{harvnb|Weiss|1976}}, p. 115</ref>
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| ::<math>\sum_{i=0}^\infty (|G_i|-1).</math>
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| ==Local computation==
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| The different may be defined for an extension of local fields ''L'' / ''K''. In this case we may take the extension to be [[Simple extension|simple]], generated by a primitive element α which also generates a [[power integral basis]]. If ''f'' is the minimal polynomial for α then the different is generated by ''f'''(α).
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{Bourbaki EHM}}
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| * {{Citation | last=Dedekind | first=Richard | author-link=Richard Dedekind | title=Über die Discriminanten endlicher Körper | url=http://gdz.sub.uni-goettingen.de/en/dms/load/img/?IDDOC=39145 | year=1882 | journal=Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen | volume=29 | pages=1–56 | issue=2 }}. Retrieved 5 August 2009
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| * {{Citation | last= Fröhlich | first = Albrecht | authorlink= Albrecht Fröhlich | last2=Taylor | first2=Martin | authorlink2= Martin J. Taylor | title=Algebraic number theory | publisher=[[Cambridge University Press]] | series=Cambridge Studies in Advanced Mathematics | isbn=0-521-36664-X | year=1991 | volume=27 | zbl=0744.11001 }}
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| * {{citation| last=Hecke | first=Erich | authorlink= | title=Lectures on the theory of algebraic numbers | others=Translated by George U. Brauer and Jay R. Goldman with the assistance of R. Kotzen | series=[[Graduate Texts in Mathematics]] | volume=77 | location=New York–Heidelberg–Berlin | publisher=[[Springer-Verlag]] | year=1981 | isbn=3-540-90595-2 | zbl=0504.12001 }}
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| * {{citation | last=Narkiewicz | first=Władysław | title=Elementary and analytic theory of algebraic numbers | edition=2nd, substantially revised and extended | publisher=[[Springer-Verlag]]; [[Polish Scientific Publishers PWN|PWN-Polish Scientific Publishers]] | year=1990 | isbn=3-540-51250-0 | zbl=0717.11045 }}
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| * {{Neukirch ANT}}
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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 }}
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| * {{citation | last=Weiss | first=Edwin | title=Algebraic Number Theory | publisher=[[Chelsea Publishing]] | edition=2nd unaltered | year=1976 | isbn=0-8284-0293-0 | zbl=0348.12101 }}
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| [[Category:Algebraic number theory]]
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