Trace pairing

For a finite field extension $L/K$, the trace pairing is the bilinear formIt is nondegenerate exactly when $L/K$ is separable.

For an $R$-lattice $M$ in $L$, its trace-dual lattice isFor the integral closure $S$ of $R$, the trace dual $S^{\vee }$ is the inverse different.

For the integral closure $S$ of a Dedekind domain $R$ in a finite separable extension, the discriminant ideal is locally generated bywhere $({\alpha }_i)$ is a local $R$-basis of $S$. Intrinsically it is the image of $({\det }_{R}S{)}^{\otimes 2}$ under the determinant of the trace pairing.