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 "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 "O""K". By definition, the different ideal δ"K" is the inverse fractional ideal "I"−1: it is an ideal of "O""K".