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 $$\ell/k$$ and Galois group $$G$$. Then the following are equivalent.
 * (i) $$L/K$$ is unramified.
 * (ii) $$\mathcal{O}_L / \mathfrak{p}\mathcal{O}_L $$ is a field, where $$\mathfrak{p}$$ is the maximal ideal of $$\mathcal{O}_K$$.
 * (iii) $$[L : K] = [\ell : k]$$
 * (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 $$\operatorname{Gal}(\ell/k)$$, 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.
 * $$L = K[\pi]$$ where $$\pi$$ is a root of an Eisenstein polynomial.
 * The norm $$N(L/K)$$ contains a uniformizer of $$K$$.