Ramification 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 $\ell/k$ and Galois group $G$. Then the following are equivalent.
*(i) $L/K$ is unramified.
*(ii) $\mathcal_L / \mathfrak\mathcal_L$ is a field, where $\mathfrak$ is the maximal ideal of $\mathcal_K$.
*(iii) : K= ell : k/math>
*(iv) The inertia subgroup of $G$ is trivial.
*(v) If $\pi$ is a uniformizing element of $K$, then $\pi$ is also a uniformizing element of $L$.

When $L/K$ is unramified, by (iv) (or (iii)), ''G'' can be identified with $\operatorname(\ell/k)$, 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 pi/math> where $\pi$ 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

*
* {{cite book , last=Weiss , first=Edwin , title=Algebraic Number Theory , publisher= Chelsea Publishing , edition=2nd unaltered , year=1976 , isbn=0-8284-0293-0 , zbl=0348.12101 , url=https://books.google.com/books?id=S38pAQAAMAAJ&q=%22finite+extension%22

Algebraic number theory