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