Submission date: 1 March 2019

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 rm NTP_2 division rings of characteristic p, using results of Chernikov, Kaplan and Simon. We also highlight consequences of our proofs that concern NIP or simple difference fields.

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Full text arXiv 1903.00442: pdf, ps.