$\rm NIP$, and ${\rm NTP}_2$ division rings of prime characteristic
Résumé
Combining a characterisation by Bélair, 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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