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.
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]).
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).
for the valuation, the ring of integers and its maximal ideal for
As a consequence of Hensel's lemma, one can write
[3] (This is stronger than the primitive element theorem.)
are normal by (i) and trivial for sufficiently large
For the lowest indices, it is customary to call
because of its relation to splitting of prime ideals, while
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
(The map actually does not depend on the choice of the uniformizer.
[10] In the terminology of Lazard, this can be understood to mean the Lie algebra
-th primitive root of unity, can be described explicitly:[11] where e is chosen such that
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.
Various methods show that the Galois group of K is
is then called the v-th ramification group in upper numbering.
The upper numbering is defined so as to be compatible with passage to quotients:[14] if
, then (whereas lower numbering is compatible with passage to subgroups.)
Herbrand's theorem states that the ramification groups in the lower numbering satisfy
), and that the ramification groups in the upper numbering satisfy
[15][16] 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.
[17] The upper numbering is compatible with the filtration of the norm residue group by the unit groups under the Artin isomorphism.