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

Abstract

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

15 pages

References in corpus (1)

• On dp-minimal fields