2014/02/03 by Eliyahu Matzri, Matzri, Eliyahu
Mathematics · #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA
paper · pdf · doi:10.48550/arxiv.1402.0332
arxiv created 2014/02/03 · arxiv updated 2014/02/04
We bound the symbol length of elements in the Brauer group of a field K containing a Cm field (for example any field containing an algebraically closed field or a finite field), and solve the local exponent-index problem for a Cm field F. In particular, for a Cm field F, we show that every F central simple algebra of exponent pt is similar to the tensor product of at most len(pt,F)≤ t(pm-1-1) symbol algebras of degree pt. We then use this bound on the symbol length to show that the index of such algebras is bounded by (pt)^(pm-1-1), which in turn gives a bound for any algebra of exponent n via the primary decomposition. Finally for a field K containing a Cm field F, we show that every F central simple algebra of exponent pt and degree ps is similar to the tensor product of at most len(pt,ps,K)≤ len(pt,L) symbol algebras of degree pt, where L is a Cm+edL(A)+ps-t-1 field.