$\mathrm{N}\mathrm{I}\mathrm{P}$, and ${\mathrm{N}\mathrm{T}\mathrm{P}}_2$ division rings of prime characteristic

Combining a characterisation by B'elair, Kaplan, Scanlon and Wagner of certain $\mathrm{N}\mathrm{I}\mathrm{P}$ valued fields of characteristic $p$ with Dickson’s construction of cyclic algebras, we provide examples of noncommutative $\rm NIP$divisionringofcharacteristic$p$andshowthatan$\rm NIP$ division ring of characteristic $p$ has finite dimension over its centre, in the spirit of Kaplan and Scanlon’s proof that infinite $\mathrm{N}\mathrm{I}\mathrm{P}$ fields have no Artin-Schreier extension. The result extends to ${\mathrm{N}\mathrm{T}\mathrm{P}}_2$ division rings of characteristic $p$, using results of Chernikov, Kaplan and Simon. We also highlight consequences of our proofs that concern $\mathrm{N}\mathrm{I}\mathrm{P}$ or simple difference fields.