Cyclic and non-cyclic division algebras of finite dp-rank

TL;DR

This paper constructs examples of finite dp-rank division algebras, including non-cyclic ones, answering a question about their existence and providing new insights into their structure over ultraproducts of p-adic numbers.

Contribution

It demonstrates the existence of division algebras with finite dp-rank, including non-cyclic cases, and explores their properties over ultraproducts of p-adic fields.

Findings

Existence of division algebras with dp-rank equal to the square of a prime.

Construction of non-cyclic finite dp-rank division algebras.

Examples over ultraproducts of p-adic numbers.

Abstract

Milliet asks the following question: given two prime numbers $p\neq q$, is there a division algebra of characteristic $p$ which is of dp-rank $q^2$ and of dimension $q^2$ over its center? We answer in the affirmative. We also give an example of a finite burden central division algebra over some ultraproduct of $p$-adic numbers. As a conclusion we revisit an example of Albert to prove that there exists non-cyclic division algebras of finite dp-rank.