We prove that NIP valued fields of positive characteristic are henselian, and we begin to generalize the known results on dp-minimal fields to dp-finite fields. On any unstable dp-finite field K, we define a type-definable group of “infinitesimals,” corresponding to a canonical group topology on $(K,+)$. We reduce the classification of positive characteristic dp-finite fields to the construction of non-trivial $\mathrm{Aut}(K/A)$-invariant valuation rings.

中文翻译：

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Dp有限域I（A）：无穷小

我们证明具有正特性的NIP值字段是henselian，并且我们开始将dp最小字段的已知结果推广到dp有限字段。在任何不稳定的dp有限域K上，我们定义一个类型可定义的“无穷小”组，对应于上的规范组拓扑$（ķ，+）$。我们将正特征dp有限域的分类简化为非平凡的构造$\mathrm{ut}（ķ/\mathrm{一个}）$-不变估值环。