关于极值域和驯服值域的注记
Notes on extremal and tame valued fields
摘要：
本文将\cite{[AKP]}中极值域的刻画推广到具有完美剩余域的混合特征值域缺失的情形。这导致了一个完整的特征驯服值领域是极值。证明的关键是一个关于混合特征驯服值域的模型论结果。进一步证明了在有限$p$度的极值域上，所有可加多项式的象都具有最优逼近性质.这一事实可以用来改进有限域上的洛朗级数场的初等理论的\cite{[K1]}中提出的公理系统。最后，我们给出了一些例子来说明当我们试图刻画具有不完美剩余域的极值域时所面临的问题。为此，我们描述了几种方法来构建极值字段，特别是，我们表明，在每一个$\aleph_1$饱和值字段的估值是一个组成的秩1的极值估值。
英文摘要：
We extend the characterization of extremal valued fields given in \cite{[AKP]} to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finite $p$-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in \cite{[K1]} for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every $\aleph_1$ saturated valued field the valuation is a composition of extremal valuations of rank 1.
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