Ramification theory of valuations
In mathematics, the ramification theory of valuations studies the set of extensions of a valuation v of a field K to an extension L of K. It is a generalization of the ramification theory of Dedekind domains.The structure of the set of extensions is known better when L/K is Galois.Let be a valued field and let L be a finite Galois extension of K. Let Sv be the set of equivalence classes of extensions of v to L and let G be the Galois group of L over K. Then G acts on Sv by σ = . In fact, this action is transitive.
Given a fixed extension w of v to L, the decomposition group of w is the stabilizer subgroup Gw of , i.e. it is the subgroup of G consisting of all elements that fix the equivalence class ∈ Sv.
Let mw denote the maximal ideal of w inside the valuation ring Rw of w. The inertia group of w is the subgroup Iw of Gw consisting of elements σ such that σx ≡ x for all x in Rw. In other words, Iw consists of the elements of the decomposition group that act trivially on the residue field of w. It is a normal subgroup of Gw.
The reduced ramification index e is independent of w and is denoted e. Similarly, the relative degree f is also independent of w and is denoted f.
