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 ${\mathcal {D}}_{B/A}$ 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 of local fields. Then ${\mathfrak {p}}$ is unramified $\Leftrightarrow B/pB$ is a field $\Leftrightarrow$ the ramification index of ${\mathfrak {q}}$ over ${\mathfrak {p}}$ is 1.