Different ideal

If OK 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{tr}~x^2 is an integral quadratic form on OK. 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 OK. By definition, the different ideal δK is the inverse fractional ideal I−1: it is an ideal of OK. The ideal norm of δK is equal to the ideal of Z generated by the field discriminant DK 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): The different ideal is generated by the differents of all integers α in OK. 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 δL / 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 ΔL / K. In a tower of fields L / K / F the relative differents are related by δL / F = δL / ''K'δ'K / F''. The relative different equals the annihilator of the relative Kähler differential module \Omega^1_{O_L/O_K}: : \delta_{L/K} = \{ x \in O_L : x \mathrm{d} y = 0 \text{ for all } y \in O_L \} . The ideal class of the relative different δL / K is always a square in the class group of OL, 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 OK: indeed, it is the square of the Steinitz class for OL as a OK-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 ΔL / K. More precisely, if :p = P1e(1) ... ''P'k'e(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 + eνP(e) − 1. The differential exponent can be computed from the orders of the higher ramification groups for Galois extensions: \sum_{i=0}^\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==