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Abstract: We classify Artin-Schreier extensions of valued fields with non-trivialdefect according to whether they are connected with purely inseparableextensions with non-trivial defect, or not. We use this classification to showthat in positive characteristic, a valued field is algebraically complete ifand only if it has no proper immediate algebraic extension and every finitepurely inseparable extension is defectless. This result is an important toolfor the construction of algebraically complete fields. We also considerextremal fields = fields for which the values of the elements in the images ofarbitrary polynomials always assume a maximum. We characterize inseparablydefectless, algebraically maximal and separable-algebraically maximal fields interms of extremality, restricted to certain classes of polynomials. We give asecond characterization of algebraically complete fields, in terms of theircompletion. Finally, a variety of examples for Artin-Schreier extensions ofvalued fields with non-trivial defect is presented.



Author: Franz-Viktor Kuhlmann

Source: https://arxiv.org/







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