Ramification Group - Ramification Groups in Lower Numbering

Ramification Groups in Lower Numbering

Let be a finite Galois extension of local fields with group and finite residue fields . We shall write for the valuation, the ring of integers and its maximal ideal for . It is known that one can write for some where is the ring of integers of . (This is stronger than the primitive element theorem and is a consequence of Hensel's lemma.) 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)

(i) shows that are normal and (ii) shows that for sufficiently large . is then called the -th ramification group, and they form a finite decreasing filtration of with . is called the inertia subgroup of . Note:

• unramified.
• tamely ramified (i.e., the ramification index is prime to the residue characteristic.)

The study of ramification groups reduces to the totally ramified case since one has for .

One also defines the function . (ii) in the above shows is independent of choice of and, moreover, the study of the filtration is essentially equivalent to that of . satisfies the following: for ,

Fix a uniformizer of . then induces the injection where . It follows from this

• is cyclic of order prime to
• is a product of cyclic groups of order .

In particular, is a p-group and is solvable.

The ramification groups can be used to compute the different of the extension and that of subextensions:

If is a normal subgroup of, then, for, .

Combining this with the above one obtains: for a subextension corresponding to ,

If, then . In the terminology of Lazard, this can be understood to mean the Lie algebra is abelian.