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.

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.

Decomposition group and inertia group

Let (K, v) 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[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 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 Gw of [w], i.e. it is the subgroup of G consisting of all elements that fix the equivalence class [w] ∈ 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 (mod mw) 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(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 in lower numbering

Ramification groups are a refinement of the Galois group G of a finite L/K Galois extension of local fields. We shall write for the valuation, the ring of integers and its maximal ideal for L. As a consequence of Hensel's lemma, one can write for some where is the ring of integers of K. (This is stronger than the primitive element theorem.) Then, for each integer i \ge -1, we define G_i to be the set of all s \in G that satisfies the following equivalent conditions. The group G_i is called i-th ramification group. They form a decreasing filtration, In fact, the G_i are normal by (i) and trivial for sufficiently large i by (iii). For the lowest indices, it is customary to call G_0 the inertia subgroup of G because of its relation to splitting of prime ideals, while G_1 the wild inertia subgroup of G. The quotient G_0 / G_1 is called the tame quotient. The Galois group G and its subgroups G_i are studied by employing the above filtration or, more specifically, the corresponding quotients. In particular, The study of ramification groups reduces to the totally ramified case since one has for i \ge 0. One also defines the function. (ii) in the above shows i_G is independent of choice of \alpha and, moreover, the study of the filtration G_i is essentially equivalent to that of i_G. i_G satisfies the following: for s, t \in G, Fix a uniformizer \pi of L. Then induces the injection where. (The map actually does not depend on the choice of the uniformizer. ) It follows from this In particular, G_1 is a p-group and G_0 is solvable. The ramification groups can be used to compute the different of the extension L/K and that of subextensions: If H is a normal subgroup of G, then, for,. Combining this with the above one obtains: for a subextension F/K corresponding to H, If, then. In the terminology of Lazard, this can be understood to mean the Lie algebra is abelian.

Example: the cyclotomic extension

The ramification groups for a cyclotomic extension, where \zeta is a p^n-th primitive root of unity, can be described explicitly: where e is chosen such that.

Example: a quartic extension

Let K be the extension of Q2 generated by. The conjugates of x_1 are, x_3 = -x_1, x_4 = -x_2. A little computation shows that the quotient of any two of these is a unit. Hence they all generate the same ideal; call it π. \sqrt{2} generates π2; (2)=π4. Now, which is in π5. and which is in π3. Various methods show that the Galois group of K is C_4, cyclic of order 4. Also: and so that the different x_1 satisfies X4 − 4X2 + 2, which has discriminant 2048 = 211.

Ramification groups in upper numbering

If u is a real number \ge -1, let G_u denote G_i where i the least integer \ge u. In other words, Define \phi by where, by convention, (G_0 : G_t) is equal to if t = -1 and is equal to 1 for. Then \phi(u) = u for. It is immediate that \phi is continuous and strictly increasing, and thus has the continuous inverse function \psi defined on. Define . G^v is then called the v-th ramification group in upper numbering. In other words,. Note. The upper numbering is defined so as to be compatible with passage to quotients: if H is normal in G, then (whereas lower numbering is compatible with passage to subgroups.)

Herbrand's theorem

Herbrand's theorem states that the ramification groups in the lower numbering satisfy (for where L/F is the subextension corresponding to H), and that the ramification groups in the upper numbering satisfy. This allows one to define ramification groups in the upper numbering for infinite Galois extensions (such as the absolute Galois group of a local field) from the inverse system of ramification groups for finite subextensions. The upper numbering for an abelian extension is important because of the Hasse–Arf theorem. It states that if G is abelian, then the jumps in the filtration G^v are integers; i.e., whenever \phi(i) is not an integer. The upper numbering is compatible with the filtration of the norm residue group by the unit groups under the Artin isomorphism. The image of G^n(L/K) under the isomorphism is just