Idea: from number theory. Take field extension \(L/k\) and write \(G = { \mathsf{Gal}} (L_{/ {k}} )\). Define the inertia subgroup \(I\leq G\). Then \(L\) is
-
Ramified iff \(I = \emptyset\).
- Equivalently, \({\mathcal{O}}_L/{\mathfrak{m}}_{{\mathcal{O}}_k} {\mathcal{O}}_L\) is a field
- Totally unramified iff \(I = G\)