Let $K$ be an NIP field and let $v$ be a Henselian valuation on $K$. We ask whether $(K,v)$ is NIP as a valued field. By a result of Shelah, we know that if $v$ is externally definable, then $(K,v)$ is NIP. Using the definability of the canonical $p$-Henselian valuation, we show that whenever the residue field of $v$ is not separably closed, then $v$ is externally definable. In the case of separably closed residue field, we show that $(K,v)$ is NIP as a pure valued field.

中文翻译：

---

NIP 油田的 Henselian 扩展

让$钾$是一个 NIP 领域，让$v$是 Henselian 估值$钾$. 我们问是否$(钾,v)$是 NIP 作为一个有价值的领域。通过 Shelah 的结果，我们知道如果$v$是外部可定义的，那么$(钾,v)$是 NIP。使用规范的可定义性$p$-Henselian valuation，我们表明每当剩余字段$v$不是可分离封闭的，那么$v$是外部可定义的。在可分离封闭的剩余场的情况下，我们表明$(钾,v)$NIP 是一个纯值域。