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.
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.