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 (K, v) be a valued field and let L be a finite Galois extension of K. Let S_v 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 S_v by σ[w] = [w ○ σ] (i.e. w is a representative of the equivalence class [w] ∈ S_v and [w] is sent to the equivalence class of the composition of w with the automorphism σ : L → L; this is independent of the choice of w in [w]). In fact, this action is transitive.

Given a fixed extension w of v to L, the decomposition group of w is the stabilizer subgroup G_w of [w], i.e. it is the subgroup of G consisting of all elements that fix the equivalence class [w] ∈ S_v.

Let m_w denote the maximal ideal of w inside the valuation ring R_w of w. The inertia group of w is the subgroup I_w of G_w consisting of elements σ such that σx ≡ x (mod m_w) for all x in R_w. In other words, I_w consists of the elements of the decomposition group that act trivially on the residue field of w. It is a normal subgroup of G_w.

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