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.

• (i) $L/K$ is unramified.
• (ii) $\mathcal{O}_L / \mathcal{O}_L \mathfrak{p}$ is a field, where $\mathfrak{p}$ is the maximal ideal of $\mathcal{O}_K$.
• (iii) $[L : K] = [l : k]$
• (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 $\operatorname{Gal}(l/k)$, 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.
• $L = K[\pi]$ where $\pi$ is a root of an Eisenstein polynomial.
• The norm $N(L/K)$ contains a uniformizer of $K$.

See also

• Abhyankar's lemma