Henselian expansions of NIP fields. (English) Zbl 07845700
Summary: 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.
MSC:
| 12L12 | Model theory of fields |
| 03C45 | Classification theory, stability, and related concepts in model theory |
| 12J10 | Valued fields |
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