Finite extensions of local fields

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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[edit]

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[edit]

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[edit]

References[edit]