Different ideal

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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 all 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.