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arXiv · 1903.00442

$\rm NIP$, and ${\rm NTP}_2$ division rings of prime characteristic

Cédric Milliet
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Abstract

Combining a characterisation by B\'elair, Kaplan, Scanlon and Wagner of certain $\rm NIP$ valued fields of characteristic $p$ with Dickson's construction of cyclic algebras, we provide examples of noncommutative $\rm NIP$ division ring of characteristic $p$ and show that an $\rm NIP$ division ring of characteristic $p$ has finite dimension over its centre, in the spirit of Kaplan and Scanlon's proof that infinite $\rm NIP$ fields have no Artin-Schreier extension. The result extends to ${\rm NTP}_2$ division rings of characteristic $p$, using results of Chernikov, Kaplan and Simon. We also highlight consequences of our proofs that concern $\rm NIP$ or simple difference fields.

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Cédric Milliet. 2019-03-01. $\rm NIP$, and ${\rm NTP}_2$ division rings of prime characteristic. https://arxiv.org/abs/1903.00442

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