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 ${\displaystyle L/K}$ be a finite Galois extension of nonarchimedean local fields with finite residue fields ${\displaystyle \ell /k}$ and Galois group ${\displaystyle G}$. Then the following are equivalent.

• (i) ${\displaystyle L/K}$ is unramified.
• (ii) ${\displaystyle {\mathcal {O}}_{L}/{\mathcal {O}}_{L}{\mathfrak {p}}}$ is a field, where ${\displaystyle {\mathfrak {p}}}$ is the maximal ideal of ${\displaystyle {\mathcal {O}}_{K}}$.
• (iii) ${\displaystyle [L:K]=[\ell :k]}$
• (iv) The inertia subgroup of ${\displaystyle G}$ is trivial.
• (v) If ${\displaystyle \pi }$ is a uniformizing element of ${\displaystyle K}$, then ${\displaystyle \pi }$ is also a uniformizing element of ${\displaystyle L}$.

When ${\displaystyle L/K}$ is unramified, by (iv) (or (iii)), G can be identified with ${\displaystyle \operatorname {Gal} (\ell /k)}$, 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 ${\displaystyle L/K}$ be a finite Galois extension of nonarchimedean local fields with finite residue fields ${\displaystyle l/k}$ and Galois group ${\displaystyle G}$. The following are equivalent.

• ${\displaystyle L/K}$ is totally ramified
• ${\displaystyle G}$ coincides with its inertia subgroup.
• ${\displaystyle L=K[\pi ]}$ where ${\displaystyle \pi }$ is a root of an Eisenstein polynomial.
• The norm ${\displaystyle N(L/K)}$ contains a uniformizer of ${\displaystyle K}$.