A lower bound on the essential dimension of simple algebras

Abstract

Let $p$ be a prime integer and $F$ a field of characteristic different from $p$. We prove that the essential $p$-dimension ed${}_p\left(\right.CSA\left(p^r\right)\left)\right.$ of the class $CSA\left(p^r\right)$ of central simple algebras of degree $p^r$ is at least $\left(r-1\right)p^r+1$. The integer ${ed}_p\left(\right.CSA\left(p^r\right)\left)\right.$ measures complexity of the class of central simple algebras of degree $p^r$ over field extensions of $F$.