We partially generalize the known results on dp-minimal fields to dp-finite fields. We prove a dichotomy: if K is a sufficiently saturated dp-finite expansion of a field, then either K has finite Morley rank or K has a non-trivial $\mathrm{Aut}(K/A)$-invariant valuation ring for some small set A. In the positive characteristic case, we can even obtain a henselian valuation ring. Using this, we classify the positive characteristic dp-finite pure fields.

中文翻译：

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Dp有限域I（B）：正特性

我们将dp最小字段上的已知结果部分推广到dp有限字段。我们证明了一个二分法：如果K是一个场的足够饱和的dp-有限展开，则K的有限Morley秩或K的非平凡$\mathrm{ut}（ķ/\mathrm{一个}）$一些小集合A的不变估值环。在积极的特征情况下，我们甚至可以得到一个henselian估值环。使用此方法，我们对正特征dp有限纯场进行分类。