Ramification group

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 groups in lower numbering

Ramification groups are a refinement of the Galois group of a finite Galois extension of local fields. We shall write for the valuation, the ring of integers and its maximal ideal for . As a consequence of Hensel's lemma, one can write for some where is the ring of integers of . (This is stronger than the primitive element theorem.) Then, for each integer , we define to be the set of all that satisfies the following equivalent conditions.

(i) operates trivially on

(ii) for all

(iii)

The group is called -th ramification group. They form a decreasing filtration,

In fact, the are normal by (i) and trivial for sufficiently large by (iii). For the lowest indices, it is customary to call the inertia subgroup of because of its relation to splitting of prime ideals, while the wild inertia subgroup of . The quotient is called the tame quotient.