Definition of discriminant of local field Let $K$ be a local field, and $L$ be an finite extension of $K$.
Then, $L$ is also local field.
Then, what is the definition of discriminant of extension L/K ?
Discriminant of extension of $\mathbb Q$ is well-know, but I couldn't find anywhere the definition of discriminant of extension of local field.
Reference(webpage, etc・・・) is also appreciated, thank you.
 A: Question: "I couldn't find titled definition of discriminant in Neukirch, if possible, could you give me self-contained explanation here?"
Answer: You find it in the index-list.
Example: Number fields: Let $K$ be a number field with ring of integers $A:=\mathcal{O}_K\subseteq K$ and let $I \subseteq K$ be an $A$-module. Then $I$ has a $\mathbb{Z}$-basis $I\cong \mathbb{Z}\{\alpha_1,..,\alpha_n\}$ (it is a free $\mathbb{Z}$-module of rank $n$ on this basis) and you define the "discriminant" $d(I)$ of the $A$-module $I$ as follows:
$$d(I):=d(\alpha_i):= det( \sigma_j(\alpha_i))^2,$$
where $\sigma_j: K \rightarrow \overline{\mathbb{Q}}$ are the embeddings of $K$ into $\overline{\mathbb{Q}}$. The element $d(I)$ is independent wrto choice of basis $\alpha_i$ for $I$. You find the notion "different" in the index list as well. This is defined in terms of the module of Kahler differentials $\Omega^1_{A}$ as $A$-module. A similar construction defines the discriminant ideal $d(L/K) \subseteq \mathcal{O}_K$ for any finite separable extension $K \subseteq L$ of fields (page 201). You view $\mathcal{O}_L$ as a left $\mathcal{O}_K$-module. If the extension is inseparable you must use "other methods". With the above definition if $K \subseteq  L$ is inseparable it follows $d(L/K)=0$.
