Discriminant ideal ($\mathrm{disc}(S/R)$)

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.

The relative discriminant ${\mathfrak{d}}_{L/K}$ of a finite extension of number fields is the discriminant ideal of ${\mathcal{O}}_L$ over ${\mathcal{O}}_K$. It satisfiesand its prime divisors are exactly the ramified finite primes.

The relative discriminant of a finite extension of number fields is the relative norm of a fractional ideal applied to its different ideal. Consequently, at $\mathfrak{p}$,where $f_{\mathfrak{P}/\mathfrak{p}}$ is the residue-field degree and $d_{\mathfrak{P}}$ is the different exponent. Over $\mathbb{Q}$, $N({\mathfrak{D}}_{E/\mathbb{Q}})=|D_E|$.