# 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

## References

- Cassels, J.W.S. (1986).
*Local Fields*. London Mathematical Society Student Texts.**3**. Cambridge University Press. ISBN 0-521-31525-5. Zbl 0595.12006. - Weiss, Edwin (1976).
*Algebraic Number Theory*(2nd unaltered ed.). Chelsea Publishing. ISBN 0-8284-0293-0. Zbl 0348.12101.