Graded and Valued Field Extensions

Abstract

We pursue in this chapter the investigation of valuations through the graded structures associated to the valuation filtration. The graded field of a valued field is an enhanced version of the residue field, inasmuch as it encapsulates information about the value group in addition to the residue field. It thus captures much of the structure of the field, particularly in the Henselian case. This point is made clear in §5.2, where we show that—when the ramification is tame—Galois groups and their inertia subgroups of Galois extensions of valued fields can be determined from the corresponding extension of graded fields. Henselian fields are shown to satisfy a tame lifting property from graded field extensions, generalizing the inertial lifting property. In §5.1, we lay the groundwork for the subsequent developments by an independent study of graded fields, their algebraic extensions and their Galois theory.