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In [[Algebraic number theory|algebraic number theory]], the '''different ideal''' (sometimes simply the '''different''') is defined to measure the (possible) lack of duality in the [[Ring of integers|ring of integers]] of an [[Algebraic number field|algebraic number field]] ''K'', with respect to the [[Field trace|field trace]].  It then encodes the [[Ramification (mathematics)#In algebraic number theory|ramification]] data for prime ideals of the ring of integers. It was introduced by [[Biography:Richard Dedekind|Richard Dedekind]] in 1882.<ref>{{harvnb|Dedekind|1882}}</ref><ref>{{harvnb|Bourbaki|1994}}, p. 102</ref>
==Definition==
If ''O''<sub>''K''</sub> is the ring of integers of ''K'', and ''tr'' denotes the field trace from ''K'' to the rational number field '''Q''', then

:<math> x \mapsto \mathrm{tr}~x^2 </math>

is an integral quadratic form on ''O''<sub>''K''</sub>. Its 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''</sub>, then ''I'' is a [[Fractional ideal|fractional ideal]] of ''K'' containing ''O''<sub>''K''</sub>. By definition, the '''different ideal''' δ<sub>''K''</sub> is the inverse fractional ideal ''I''<sup>&minus;1</sup>: it is an ideal of ''O''<sub>''K''</sub>.

The [[Ideal norm|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&nbsp;''K''.

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

:<math>\delta(\alpha) = \prod \left({\alpha - \alpha^{(i)}}\right) \ </math>

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''</sub>.<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>

The different is also defined for a [[Finite extensions of local fields|finite degree extension]] of [[Local field|local field]]s.  It plays a basic role in [[Pontryagin duality]] for p-adic fields.

==Relative different==
The '''relative different''' δ<sub>''L''&nbsp;/&nbsp;''K''</sub> is defined in a similar manner for an extension of number fields ''L''&nbsp;/&nbsp;''K''.  The relative norm of the relative different is then equal to the relative discriminant Δ<sub>''L''&nbsp;/&nbsp;''K''</sub>.<ref name=Neukirch201>{{harvnb|Neukirch|1999|p=201}}</ref>  In a [[Tower of fields|tower of fields]] ''L''&nbsp;/&nbsp;''K''&nbsp;/&nbsp;''F'' the relative differents are related by δ<sub>''L''&nbsp;/&nbsp;''F''</sub> = δ<sub>''L''&nbsp;/&nbsp;''K''</sub>''δ''<sub>''K''&nbsp;/&nbsp;''F''</sub>.<ref name=Neukirch195/><ref name=FT126>{{harvnb|Fröhlich|Taylor|1991|p=126}}</ref>

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>

<math>\delta_{L/K} = \{ x \in O_L : x \mathrm{d} y = 0 \text{ for all } y \in O_L \} . </math>

The ideal class of the relative different δ<sub>''L''&nbsp;/&nbsp;''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>

==Ramification==
The relative different encodes the [[Ramification (mathematics)#In algebraic number theory|ramification]] data of the field extension ''L''&nbsp;/&nbsp;''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''&nbsp;/&nbsp;''K''</sub>.  More precisely, if

:''p'' = ''P''<sub>1</sub><sup>''e''(1)</sup> ... ''P''<sub>''k''</sub><sup>''e''(''k'')</sup>

is the factorisation of ''p'' into prime ideals of ''L'' then ''P''<sub>''i''</sub> divides the relative different δ<sub>''L''&nbsp;/&nbsp;''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&nbsp;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''&nbsp;&minus;&nbsp;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''&nbsp;+&nbsp;''e''ν<sub>''P''</sub>(e)&nbsp;&minus;&nbsp;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>
<math display="block">\sum_{i=0}^\infty (|G_i|-1).</math>

==Local computation==
The different may be defined for an extension of local fields ''L''&nbsp;/&nbsp;''K''.  In this case we may take the extension to be 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&nbsp;''f'''(α).

==Notes==
{{reflist}}

==References==
* {{Bourbaki EHM}}
* {{Citation | last=Dedekind | first=Richard |  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
* {{Citation | last1= Fröhlich | first1 = Albrecht |  last2=Taylor | first2=Martin |  title=Algebraic number theory | publisher={{wipe|Cambridge University Press}} | series=Cambridge Studies in Advanced Mathematics | isbn=0-521-36664-X | year=1991 | volume=27 | zbl=0744.11001 }}
* {{citation| last=Hecke | first=Erich | title=Lectures on the theory of algebraic numbers | translator=George U. Brauer | translator2=Jay R. Goldman | translator3 = with the assistance of R. Kotzen | series=[[Graduate Texts in Mathematics]] | volume=77 | location=New York–Heidelberg–Berlin | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | year=1981 | isbn=3-540-90595-2 | zbl=0504.12001 }} 
* {{citation | last=Narkiewicz | first=Władysław | title=Elementary and analytic theory of algebraic numbers | edition=2nd, substantially revised and extended | publisher=[[Physics:Springer-Verlag|Springer-Verlag]]; PWN-Polish Scientific Publishers | year=1990 | isbn=3-540-51250-0 | zbl=0717.11045 | url-access=registration | url=https://archive.org/details/elementaryanalyt0000nark }}
* {{Neukirch ANT}}
* {{citation | last=Serre | first=Jean-Pierre |  title=[[Local Fields]] | translator-last1=Greenberg|translator-first=Marvin Jay|translator-link1=Marvin Greenberg | series=[[Graduate Texts in Mathematics]] | volume=67 | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 }}
* {{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 | url-access=registration | url=https://archive.org/details/algebraicnumbert00weis_0 }}

[[Category:Algebraic number theory]]

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