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 OK 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 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 codfifferent or Dedekind's complementary module as the set I of x ∈ K such that tr(xy) is an integer for all y in OK, 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 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 an finite degree extension of local fields. It plays a basic role in Pontryagin duality for p-adic fields.
Read more about Different Ideal: Relative Different, Ramification, Local Computation
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