  We classify Artin-Schreier extensions of valued fields with non-trivial
defect according to whether they are connected with purely inseparable
extensions with non-trivial 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 non-trivial defect is presented.
