inertia group

《Number Fields》 · 第 69 页 · Chapter 4「Galois Theory Applied to Prime Decomposition」开篇定义

Let $K$ and $L$ be number fields, and assume that $L$ is a normal extension of $K$. Thus the Galois group $G$, consisting of all automorphisms of $L$ which fix $K$ pointwise, has order $n=[L:K]$. As usual we let $R$ and $S$ denote the corresponding number rings. Fixing a prime $P$ of $R$, [...] for each prime $Q$ lying over $P$, we define two subgroups of $G$: The inertia group: $E=E(Q\mid P)=\{\sigma\in G:\sigma(\alpha)\equiv\alpha\pmod Q\ \forall\alpha\in S\}$.