  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.
