Asymptotic Brauer $p$-Dimension

Abstract

We define and compute ${\mathrm{ABrd}}_p(F)$, the asymptotic Brauer $p$-dimension of a field $F$, in cases where $F$ is a rational function field or Laurent series field. ${\mathrm{ABrd}}_p(F)$ 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 $F_0(\hat{I}{\pm }_1,\dots ,\hat{I}{\pm }_n)$ and $F_0((\hat{I}{\pm }_1))\dots ((\hat{I}{\pm }_n))$ where $F_0$ is a perfect field of characteristic $p>0$ when $n\ge 2$ the asymptotic Brauer $p$-dimension is $n$. We also show that it is $n-1$ when $F=F_0((\hat{I}{\pm }_1))\dots ((\hat{I}{\pm }_n))$ 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 ${\mathrm{Brd}}_p(F)=2$ that share no maximal subfields despite their tensor product being non-division.