Different ideal: Difference between revisions
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In [[algebraic number theory]], the '''different ideal''' is an invariant attached to an extension of [[algebraic number field]]s. | In [[algebraic number theory]], the '''different ideal''' is an invariant attached to an extension of [[algebraic number field]]s. | ||
Let ''L''/''K'' be such an extension, with [[ring of integers|rings of integers]] ''O''<sub>''L''</sub> and ''O''<sub>''K''</sub> respectively. The relative [[field trace|trace]] defines a bilinear on ''L'' by | Let ''L''/''K'' be such an extension, with [[ring of integers|rings of integers]] ''O''<sub>''L''</sub> and ''O''<sub>''K''</sub> respectively. The relative [[field trace|trace]] defines a [[bilinear form]] on ''L'' by | ||
:<math> (x,y) \mapsto \mathrm{tr}_{L/K} (x \cdot y) \, </math> | :<math> (x,y) \mapsto \mathrm{tr}_{L/K} (x \cdot y) \, </math> | ||
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in which the dual of ''O''<sub>''L''</sub> is a [[fractional ideal]] of ''L'' containing ''O''<sub>''L''</sub>. The ''(relative) different'' δ<sub>''L''/''K''</sub> is the inverse of this fractional ideal: it is an ideal of ''O''<sub>''L''</sub>. | in which the dual of ''O''<sub>''L''</sub> is a [[fractional ideal]] of ''L'' containing ''O''<sub>''L''</sub>. The ''(relative) different'' δ<sub>''L''/''K''</sub> is the inverse of this fractional ideal: it is an ideal of ''O''<sub>''L''</sub>. | ||
The [[relative norm]] of the relative different is equal to the relative discriminant Δ<sub>''L''/''K''</sub>. 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>. | The [[relative norm]] of the relative different is equal to the [[Discriminant of an algebraic number field|relative discriminant]] Δ<sub>''L''/''K''</sub>. 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>. | ||
==Ramification== | ==Ramification== | ||
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==Local computation== | ==Local computation== | ||
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). | 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 (field theory)|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).[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 07:01, 7 August 2024
In algebraic number theory, the different ideal is an invariant attached to an extension of algebraic number fields.
Let L/K be such an extension, with rings of integers OL and OK respectively. The relative trace defines a bilinear form on L by
in which the dual of OL is a fractional ideal of L containing OL. The (relative) different δL/K is the inverse of this fractional ideal: it is an ideal of OL.
The relative norm of the relative different is equal to the relative discriminant ΔL/K. In a tower of fields L/K/F the relative differents are related by δL/F = δL/K δK/F.
Ramification
The relative different encodes the ramification data of the field extension L/K. A prime ideal p of K ramifies in L if and only if it divides the relative discriminant ΔL/K. If
- p = P1e(1) ... Pke(k)
is the factorisation of p into prime ideals of L then Pi divides the relative different δL/K if and only if Pi is ramified, that is, if and only if the ramification index e(i) is greater than 1. 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. In the case when P is wildly ramified the differential exponent lies in the range e to e+νP(e)-1.
Local computation
The different may be defined for an extension of local fields L/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 f'(α).