\author{Dave Benson}

\address
{Institute of Mathematics \\
University of Aberdeen \\
King's College \\
Aberdeen AB24 3UE \\
Scotland, UK}

\author[Zinovy Reichstein]{Zinovy Reichstein}

\address
{Department of Mathematics \\
University of British Columbia \\
Vancouver
\\
CANADA}

\title[Fields of definition for representations]{Fields of definition 
for representations of associative algebras}

\keywords
{Modular representation, field of definition, finite representation type,
essential dimension}
\subjclass[2010]{16G10, 16G60, 20C05}

\begin{abstract}
We examine situations, where representations of a finite-dimensional
$F$-algebra $A$ defined over a separable extension field
$K/F$, have a unique minimal field of definition. Here 
the base field $F$ is assumed to be a $C_1$-field.
In particular, $F$ could be a finite field or $k(t)$ or $k((t))$,
where $k$ is algebraically closed.

We show that a unique minimal field of definition exists if
(a) $K/F$ is an algebraic extension or (b) $A$ is 
of finite representation type. Moreover, in these situations
the minimal field of definition is a finite extension of $F$.
This is not the case if $A$ is of infinite representation type 
or $F$ fails to be $C_1$.
As a consequence, we compute the essential dimension of the functor
of representations of a finite group, generalizing a theorem
of N.~Karpenko, J.~Pevtsova and the second author.
\end{abstract}