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A classification of Artin-Schreier defect extensions and characterizations of defectless fields. (English) Zbl 1234.12007
Ill. J. Math. 54, No. 2, 397-448 (2010); correction ibid. 63, No. 3, 463-468 (2019).
The paper is well written with lucid exposition. The author gives a complete classification of Artin-Schreier extensions of valued fields with non trivial defect. This classification has been used to show that a henselian valued field \((K,v)\) of positive characteristic is defectless if and only if it has no proper immediate algebraic extension and its every finite purely inseparable extension is defectless. The author also gives a new characterization of valued fields satisfying the latter property as well as of algebraically maximal and separable algebraically maximal fields in terms of extremality, restricted to certain classes of polynomials. It is also proved that a henselian valued field of positive characteristic is defectless if and only if its completion is a defectless field. The paper gives some easy proofs of the results proved in F. Delon’s thesis. A characterization of algebraically maximal valued fields in terms of extremality and that of defectless extensions using saturated distinguished chains is given in [A. Bishnoi and the reviewer, “On algebraically maximal valued fields and defectless extensions,” Can. Math. Bull. (to appear), see doi:10.4153/CMB-2011-148-0].

MSC:
12J10 Valued fields
12J25 Non-Archimedean valued fields
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Full Text: Euclid
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