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.

Introduction

To motivate the below, we start with how one reduces a global case to a local case. Thus, let A be a Dedekind domain with the field of fractions K and $L/K$ a finite separable extension. Let ${\mathfrak {q}}\in \operatorname {Spec} B$ and ${\mathfrak {p}}=A\cap {\mathfrak {q}}$ . A basic result in algebraic number theory is that the extension $L/K$ is unramified at ${\mathfrak {q}}$ if and only if ${\mathfrak {q}}$ does not divide the different ${\mathfrak {D}}_{B/A}$ of $B$ over $A$ . (Upon taking a norm, this says that $L/K$ is unramified at ${\mathfrak {q}}\cap A$ if and only if ${\mathfrak {q}}\cap A$ does not divide the discriminant of $B$ over $A$ .) Since the different commutes with localization and completion, this reduces to the case when $A,B$ are complete; i.e., $L,K$ are local fields.

Let $L/K$ be a finite Galois extension of nonarchimedean local fields with finite residue fields $l,k$ and group G. The preceding applies with $B={\mathcal {O}}_{L}$ and $A={\mathcal {O}}_{K}.$

Unramified extension

The following are equivalent.

(i) $L/K$ is unramified.
(ii) ${\mathcal {O}}_{L}/{\mathfrak {q}}$ is a separable "field" extension of ${\mathcal {O}}_{K}/{\mathfrak {p}}$ .
(iii) $[L:K]=[l:k]$
(iv) The inertia subgroup of G is trivial.
(v) If $\pi$ is a uniformizing element of $L$ , then $\pi$ is also a uniformizing element of $K$ .