A classification of Artin–Schreier defect extensions and characterizations of defectless fields

Abstract

We classify Artin–Schreier extensions of valued fields with nontrivial defect according to whether they are connected with purely inseparable extensions with nontrivial defect, or not. We use this classification to show that in positive characteristic, a valued field is algebraically complete if and only if it has no proper immediate algebraic extension and every finite purely inseparable extension is defectless. This result is an important tool for the construction of algebraically complete fields. We also consider extremal fields (=fields for which the values of the elements in the images of arbitrary polynomials always assume a maximum). We characterize inseparably defectless, algebraically maximal and separable-algebraically maximal fields in terms of extremality, restricted to certain classes of polynomials. We give a second characterization of algebraically complete fields, in terms of their completion. Finally, a variety of examples for Artin–Schreier extensions of valued fields with nontrivial defect is presented.