Artin-Schreier缺陷扩张的分类及无缺陷域的刻画
A classification of Artin-Schreier defect extensions and a characterization of defectless fields
摘要：
根据与非平凡亏值域的纯不可分扩张是否连通，对非平凡亏值域的Artin-Schreier扩张进行分类.我们利用这一分类证明了在正特征下，一个值域是代数完备的当且仅当它没有真的直接代数扩张，并且每个有限的纯不可分扩张都是无缺陷的.这个结果是构造代数完全域的重要工具。我们还考虑了极值场（=任意多项式的象中元素的值总是取最大值的场）。我们刻画了不可分离的无缺陷，代数极大和可分性代数极大领域的极值，仅限于某些类的多项式。我们给出了代数完全域的第二个特征，根据他们的完成。最后，给出了具有非平凡亏损的值域的Artin-Schreier扩张的各种例子。
英文摘要：
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.
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