Essential Dimension, Symbol Length and -rank

Abstract

We prove that the essential dimension of central simple algebras of degree p ℓm and exponent p m over fields of characteristic p is at least ℓ + 1. We do this by observing that the p-rank of F bounds the symbol length in Br p m (F) and that there exist indecomposable p-algebras of degree p ℓm and exponent p m . We also prove that the symbol length of the Milne-Kato cohomology group H n+1 p m (F) is bounded from above by r n where r is the p-rank of the field, and provide upper and lower bounds for the essential dimension of Brauer classes of a given symbol length.