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 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.

When L/K is unramified, by (iv) (or (iii)), G can be identified with \operatorname{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.

Totally ramified extension

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.

See also

References

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