Asymptotic Brauer $p$-Dimension

TL;DR

This paper introduces the concept of asymptotic Brauer p-dimension for fields, computes it for certain rational and Laurent series fields, and explores its properties and examples related to cyclic algebras.

Contribution

It defines and calculates the asymptotic Brauer p-dimension for specific classes of fields, extending understanding of Brauer groups in these contexts.

Findings

For rational function and Laurent series fields over perfect fields of characteristic p, the asymptotic Brauer p-dimension equals the number of variables.

In Laurent series fields over algebraically closed fields of characteristic not p, the dimension is one less.

Examples show pairs of cyclic algebras of prime degree with no shared maximal subfields but non-division tensor products.

Abstract

We define and compute $\operatorname{ABrd}_p(F)$, the asymptotic Brauer $p$-dimension of a field $F$, in cases where $F$ is a rational function field or Laurent series field. $\operatorname{ABrd}_p(F)$ is defined like the Brauer $p$-dimension except it considers finite sets of Brauer classes instead of single classes. Our main result shows that for fields $F_0(\alpha_1,\dots,\alpha_n)$ and $F_0 (\!( \alpha_1)\!) \dots(\!(\alpha_n)\!)$ where $F_0$ is a perfect field of characteristic $p>0$ when $n \geq 2$ the asymptotic Brauer $p$-dimension is $n$. We also show that it is $n-1$ when $F=F_0 (\!( \alpha_1)\!) \dots(\!(\alpha_n)\!)$ and $F_0$ is algebraically closed of characteristic not $p$. We conclude the paper with examples of pairs of cyclic algebras of odd prime degree $p$ over a field $F$ for which $\operatorname{Brd}_p(F)=2$ that share no maximal subfields despite their tensor product being non-division.