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 L / K be a finite Galois extension of nonarchimedean local fields with finite residue fields l / k and Galois group G . Then the following are equivalent.
(i) L / K is unramified.(ii) O L / O L p is a field, where p is the maximal ideal of O K .(iii) [ L : K ] = [ l : k ] (iv) The inertia subgroup of G is trivial.(v) If π is a uniformizing element of K , then π is also a uniformizing element of L .When L / K is unramified, by (iv) (or (iii)), G can be identified with Gal ( l / k ) , 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.
Again, let L / K be a finite Galois extension of nonarchimedean local fields with finite residue fields l / k and Galois group G . The following are equivalent.
L / K is totally ramified G coincides with its inertia subgroup. L = K [ π ] where π is a root of an Eisenstein polynomial.The norm N ( L / K ) contains a uniformizer of K .