In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension.

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.

The reduced ramification index e(w/v) is independent of w and is denoted e(v). Similarly, the relative degree f(w/v) is also independent of w and is denoted f(v).

Ramification groups are a refinement of the Galois group ${\displaystyle G}$ of a finite ${\displaystyle L/K}$ Galois extension of local fields. We shall write ${\displaystyle w,{\mathcal {O}}_{L},{\mathfrak {p}}}$ for the valuation, the ring of integers and its maximal ideal for ${\displaystyle L}$. As a consequence of Hensel's lemma, one can write ${\displaystyle {\mathcal {O}}_{L}={\mathcal {O}}_{K}[\alpha ]}$ for some ${\displaystyle \alpha \in L}$ where ${\displaystyle {\mathcal {O}}_{K}}$ is the ring of integers of ${\displaystyle K}$. (This is stronger than the primitive element theorem.) Then, for each integer ${\displaystyle i\geq -1}$, we define ${\displaystyle G_{i}}$ to be the set of all ${\displaystyle s\in G}$ that satisfies the following equivalent conditions.