Essential dimension of separable algebras embedding in a fixed central simple algebra

Abstract

In this paper we fix a central simple $F$-algebra $A$ of prime power degree and consider separable algebras over extensions $K/F$, which embed in $A_K$. We study the minimal number of independent parameters, called essential dimension, needed to define these separable algebras. In case the index of $A$ does not exceed a certain bound, the task is equivalent to the problem of computing the essential dimension of the algebraic groups $({\mathbf{PGL}}_d{)}^m⋊S_m$, which is extremely difficult in general. In the other case, however, we manage to compute the exact value of the essential dimension of the given class of separable algebras, except in one case for $A$ of index 2, which we study in greater detail.