Essential dimension, symbol length and p-rank

We prove that the essential dimension of central simple algebras of degree p^ℓm and exponent p^m over fields F containing a base-field k of characteristic p is at least ℓ + 1 when k is perfect. We do this by observing that the p-rank of F bounds the symbol length in Br_pm (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 Kato-Milne cohomology group Hⁿ_p⁺_m¹(F) is bounded from above by (_n^r) 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.