Different ideal (${\mathfrak{D}}_{L/K}$)

For a finite separable extension of local fields, the different is the inverse fractional ideal of the codifferent. If ${\mathcal{O}}_L={\mathcal{O}}_K[\alpha ]$ and $f$ is the minimal polynomial of $\alpha$, then ${\mathfrak{D}}_{L/K}=(f^{\prime }(\alpha ))$.

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 $g^{\prime }(\alpha {)}^{-1}{\mathcal{O}}_L$.

For finite separable extensions $L/E/K$,For normalized valuations this says

For a finite Galois extension of local fields,