2 Lower Ramification Group
The decomposition group of \(L/K\) is the subgroup
\[ \{ \sigma \in \operatorname{Gal}(L/K)\mid \sigma (\mathcal{O}_L) \subset \mathcal{O}_L\} \]
of \(\operatorname{Gal}(L/K)\)
The lower index of \(\sigma \in \operatorname{Gal}(L/K)\) is
\[ i_{L/K}(\sigma ) := \begin{cases} \infty , & \sigma = 1,\\ , & \sigma \ne 1. \end{cases} \]
If \(\sigma \in \operatorname{Gal}(L/K)\) lies in the decomposition group of \(L/K\), then
\[ i_{L/K}(\sigma ) = \infty \iff s = 1. \]
Proof
The lower ramification groups \(\operatorname{Gal}(L/K)_{u}\) vanish for \(u\in \mathbb {Z}\) large enough.