Asymptotic Brauer p-Dimension

2021/09/24 by Chapman, Adam, McKinnie, Kelly
#16K20 16W60 #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2109.11918

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 F₀(α₁,…,α_n) and F₀ ( ( α₁) ) …( (α_n) ) where F₀ 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=F₀ ( ( α₁) ) …( (α_n) ) and F₀ 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.

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