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

  • Roland Lötscher


In this paper we fix a central simple FF-algebra AA of prime power degree and consider separable algebras over extensions K/FK/F, which embed in AKA_K. We study the minimal number of independent parameters, called essential dimension, needed to define these separable algebras. In case the index of AA does not exceed a certain bound, the task is equivalent to the problem of computing the essential dimension of the algebraic groups (PGLd)mSm(\bold{PGL}_d)^m \rtimes 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 AA of index 2, which we study in greater detail.