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

## TL;DR

This paper constructs examples of noncommutative NIP division rings of prime characteristic and proves that such rings have finite dimension over their centers, extending to NTP_2 division rings and exploring implications for difference fields.

## Contribution

It provides the first known examples of noncommutative NIP division rings of characteristic p and establishes their finite-dimensionality over the center, extending results to NTP_2 rings.

## Key findings

- Existence of noncommutative NIP division rings of characteristic p.
- NIP division rings of characteristic p have finite dimension over their centers.
- Extension of results to NTP_2 division rings and implications for difference fields.

## 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.