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. (i)  is unramified. (ii)  is a field, where  is the maximal ideal of . (iii) 
(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. 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 . See also 
Abhyankar's lemma
Unramified morphism

References

 
 

Algebraic number theory