Inverse different (${\mathfrak{D}}_{L/K}^{-1}$)

For a finite separable extension of number fields or local fields,It is also called the codifferent. If ${\mathcal{O}}_L={\mathcal{O}}_K[\alpha ]$ with minimal polynomial $g$, it equals $g^{\prime }(\alpha {)}^{-1}{\mathcal{O}}_L$.