Asymptotic Brauer 𝑝-dimension

Abstract

We define and compute the asymptotic Brauer p p -dimension of a field F F , denoted A B r d p ( F ) ABrd_p(F) , in cases where F F is a rational function field or Laurent series field. A B r d p ( F ) ABrd_p(F) is defined like the Brauer p 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 ) F_0(\alpha _1,\dots ,\alpha _n) and F 0 ( ( α 1 ) ) … ( ( α n ) ) F_0 (\!( \alpha _1)\!) \dots (\!(\alpha _n)\!) where F 0 F_0 is a perfect field of characteristic p > 0 p>0 when n ≥ 2 n \geq 2 the asymptotic Brauer p p -dimension is n n . We also show that it is n − 1 n-1 when F = F 0 ( ( α 1 ) ) … ( ( α n ) ) F=F_0 (\!( \alpha _1)\!) \dots (\!(\alpha _n)\!) and F 0 F_0 is algebraically closed of characteristic not p p . We conclude the paper with examples of pairs of cyclic algebras of odd prime degree p p over a field F F for which Brd p ⁡ ( F ) = 2 \operatorname {Brd}_p(F)=2 that share no maximal subfields despite their tensor product being non-division.