Different Ideal

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.

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

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 codfifferent 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 f′(α) if α generates the field K (and zero otherwise). 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 an finite degree extension of local fields. It plays a basic role in Pontryagin duality for p-adic fields.