Differential exponent

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 data for prime ideals of the ring of integers. It was introduced by Richard Dedekind in 1882.

Definition

If ''O''_''K'' is the ring of integers of ''K'', and ''tr'' denotes the field trace from ''K'' to the rational number field Q, then

:$x \mapsto \mathrm~x^2$

is an integral quadratic form on ''O''_''K''. 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'' or ''Dedekind's complementary module'' as the set ''I'' of ''x'' ∈ ''K'' such that tr(''xy'') is an integer for all ''y'' in ''O''_''K'', then ''I'' is a fractional ideal of ''K'' containing ''O''_''K''. By definition, the different ideal δ_''K'' is the inverse fractional ideal ''I''⁻¹: it is an ideal of ''O''_''K''.

The ideal norm of ''δ''_''K'' is equal to the ideal of ''Z'' generated by the field discriminant ''D''_''K'' of ''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): we may write

:$\delta(\alpha) = \prod \left(\right) \$

where the α^(''i'') run over all the roots of the characteristic polynomial of α other than α itself. The different ideal is generated by the differents of all integers α in ''O''_''K''. This is Dedekind's original definition.

The different is also defined for a finite degree extension of local fields. It plays a basic role in Pontryagin duality for p-adic fields.

Relative different

The relative different δ_{''L'' / ''K''} 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_{''L'' / ''K''}. In a tower of fields ''L'' / ''K'' / ''F'' the relative differents are related by δ_{''L'' / ''F''} = δ_{''L'' / ''K''}''δ''_{''K'' / ''F''}.

The relative different equals the annihilator of the relative Kähler differential module $\Omega^1_$:

: $\delta_ = \ .$

The ideal class of the relative different δ_{''L'' / ''K''} is always a square in the class group of ''O''_''L'', the ring of integers of ''L''. Since the relative discriminant is the norm of the relative different it is the square of a class in the class group of ''O''_''K'': indeed, it is the square of the Steinitz class for ''O''_''L'' as a ''O''_''K''-module.

Ramification

The relative different encodes the 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_{''L'' / ''K''}. More precisely, if

:''p'' = ''P''₁^''e''(1) ... ''P''_''k''^''e''(''k'')

is the factorisation of ''p'' into prime ideals of ''L'' then ''P''_''i'' divides the relative different δ_{''L'' / ''K''} if and only if ''P''_''i'' 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'' + ''e''ν_''P''(e) − 1. The differential exponent can be computed from the orders of the higher ramification groups for Galois extensions:, p. 115
$\sum_^\infty (, G_i, -1).$

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).

Notes

References

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* . Retrieved 5 August 2009
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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 , url-access=registration , url=https://archive.org/details/algebraicnumbert00weis_0

Algebraic number theory