#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday (2015), 443-459

** Roland Lötscher **
Essential Dimension of Separable Algebras
Embedding in a Fixed Central Simple Algebra

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 $(\PGL_d)^m
\rtimes S_m$, 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 $A$ of
index $2$, which we study in greater detail.

2010 Mathematics Subject Classification: 16W10, 16K20

Keywords and Phrases: essential dimension, central simple algebras, separable algebra, étale
algebras, non-split algebraic group

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