Finite extensions of local fields
In algebraic number theory, through completion, the study of ramification of a prime ideal can often be reduced to the case of local fields where a more detailed analysis can be carried out with the aid of tools such as ramification groups.
In this article, a local field is non-archimedean and has finite residue field.
Let be a finite Galois extension of nonarchimedean local fields with finite residue fields and Galois group . Then the following are equivalent.
- (i) is unramified.
- (ii) is a field, where is the maximal ideal of .
- (iv) The inertia subgroup of is trivial.
- (v) If is a uniformizing element of , then is also a uniformizing element of .
When is unramified, by (iv) (or (iii)), G can be identified with , which is finite cyclic.
Totally ramified extension
Again, let be a finite Galois extension of nonarchimedean local fields with finite residue fields and Galois group . The following are equivalent.
- is totally ramified
- coincides with its inertia subgroup.
- where is a root of an Eisenstein polynomial.
- The norm contains a uniformizer of .