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.

Unramified extension

Let be a finite Galois extension of nonarchimedean local fields with finite residue fields and Galois group. Then the following are equivalent.

is unramified.
is a field, where is the maximal ideal of.
The inertia subgroup of is trivial.
If is a uniformizing element of, then is also a uniformizing element of.

When is unramified, by, G can be identified with, which is finite cyclic.
The above implies that there is an equivalence of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K.

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.