Abstract
We show that any theory of tame henselian valued fields is NIP if and only if the theory of its residue field and the theory of its value group are NIP. Moreover, we show that if (K, v) is a henselian valued field of residue characteristic \(\mathrm {char}(Kv)=p\) such that if \(p>0\), depending on the characteristic of K either the degree of imperfection or the index of the pth powers is finite, then (K, v) is NIP iff Kv is NIP and v is roughly separably tame.
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Partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via CRC 878 and under Germany’s Excellence Strategy EXC 2044–390685587, ‘Mathematics Münster: Dynamics–Geometry–Structure’, ValCoMo (ANR-13-BS01-0006), NSF (Grant No. 1665491) and a Sloan fellowship.
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Jahnke, F., Simon, P. NIP henselian valued fields. Arch. Math. Logic 59, 167–178 (2020). https://doi.org/10.1007/s00153-019-00685-8
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DOI: https://doi.org/10.1007/s00153-019-00685-8