Different ideal

In mathematics, the different ideal is defined in algebraic number theory, to account for the (possible) lack of duality in the ring of integers of an algebraic number field K , with respect to the field trace.

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

: tr ( xy )

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). Defining a fractional ideal I of K as the set of x ∈ K such that tr ( xy ) is an integer for some y in O K , then I contains O K . By definition, the different ideal δ K is I −1, an ideal of O K .

The field norm of δ K is the ideal of Z generated by the discriminant D K of K .

The different may also be defined for an extension of number fields L / K (the relative different) and for local fields. It plays a basic role in Pontryagin duality for p-adic fields.