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
formula_1
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
formula_2
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 formula_3:
formula_4
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:
formula_5
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.
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