Defect extensions and a characterization of tame fields

TL;DR

This paper characterizes tame fields in relation to defectless fields, showing their equivalence under certain conditions and introducing roughly tame fields, with constructions of Galois defect extensions.

Contribution

It provides a detailed characterization of tame fields, introduces the concept of roughly tame fields, and constructs Galois defect extensions in various characteristics.

Findings

Tame fields are exactly the valued fields with all algebraic extensions defectless in equal characteristic or rank one mixed characteristic cases.

In general, tame fields form a proper subclass of valued fields with all algebraic extensions defectless.

Every algebraic extension's defectless property is characterized by its henselization being roughly tame.

Abstract

We study the relation between two important classes of valued fields: tame fields and defectless fields. We show that in the case of valued fields of equal characteristic or rank one valued fields of mixed characteristic, tame fields are exactly the valued fields for which all algebraic extensions are defectless fields. In general tame fields form a proper subclass of valued fields for which all algebraic extensions are defectless fields. We introduce a wider class of roughly tame fields and show that every algebraic extension of a given valued field is defectless if and only if its henselization is roughly tame. Proving the above results we also present constructions of Galois defect extensions in positive as well as mixed characteristic.