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.

## Contents

* 1 Unramified extension
* 2 Totally ramified extension
* 3 See also
* 4 References

## 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.

* (i) L/K is unramified.
* (ii) 𝒪L/𝔭𝒪L is a field, where 𝔭 is the maximal ideal of 𝒪K.
* (iii) [L:K]=[ℓ:k]
* (iv) The inertia subgroup of G is trivial.
* (v) If π is a uniformizing element of K, then π is also a uniformizing element of L.

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

* L/K is totally ramified
* G coincides with its inertia subgroup.
* L=K[π] where π is a root of an Eisenstein polynomial.
* The norm N(L/K) contains a uniformizer of K.

## See also

* Abhyankar's lemma
* Unramified morphism

## References

* Cassels, J.W.S. (1986). Local Fields. London Mathematical Society Student Texts. 3. Cambridge University Press. ISBN 0-521-31525-5. https://books.google.com/books?id=UY52SQnV9w4C&q="finite+extension".
* Weiss, Edwin (1976). Algebraic Number Theory (2nd unaltered ed.). Chelsea Publishing. ISBN 0-8284-0293-0. https://books.google.com/books?id=S38pAQAAMAAJ&q="finite+extension".

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