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

Abstract

Combining a characterisation by Bélair, Kaplan, Scanlon and Wagner of certain NIP valued fields of characteristic p with Dickson's construction of cyclic algebras, we provide examples of noncommutative NIP division ring of characteristic p and show that an NIP division ring of characteristic p has finite dimension over its centre, in the spirit of Kaplan and Scanlon's proof that infinite NIP fields have no Artin-Schreier extension. The result extends to NTP2 division rings of characteristic p, using Chernikov, Kaplan and Simon's [13]. We also highlight consequences of our proofs that concern NIP or simple difference fields.