Defect extensions and a characterization of tame fields

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 henselian valued fields for which all algebraic extensions are defectless fields. In general tame fields form a proper subclass of henselian 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. (c) 2023 Elsevier Inc. All rights reserved.