Topic summary
Inertia group
Extracted from the Wikipedia article Ramification group.
Decomposition group and inertia group
Let (K, v) be a valued field and let L be a finiteGalois extension of K. Let Sv be the set of equivalenceclasses of extensions of v to L and let G be the Galois group of L over K. Then G acts on Sv by σ[w] = [w ∘ σ] (i.e. w is a representative of the equivalence class [w] ∈ Sv and [w] is sent to the equivalence class of the composition of w with the automorphismσ : L → L; this is independent of the choice of w in [w]). In fact, this action is transitive.