Asymptotic Brauer $p$-Dimension

Abstract

We define and compute 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. 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 (α 1 , . . . , α n ) and F 0 ((α 1 )) . . . ((α n )) where F 0 is a perfect field of characteristic p > 0 when n 2 the asymptotic Brauer p-dimension is n. We also show that it is n − 1 when F = F 0 ((α 1 )) . . . ((α 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 Brd p (F) = 2 that share no maximal subfields despite their tensor product being non-division.