  The chief aim of this paper is to describe a procedure which, given a
$d$-dimensional absolutely irreducible matrix representation of a finite group
over a finite field $\mathbb{E}$, produces an equivalent representation such
that all matrix entries lie in a subfield $\mathbb{F}$ of $\mathbb{E}$ which is
as small as possible. The algorithm relies on a matrix version of Hilbert's
Theorem 90, and is probabilistic with expected running time ${\rm
O}(|\mathbb{E}:\mathbb{F}|d^3)$ when $|\mathbb{F}|$ is bounded. Using similar
methods we then describe an algorithm which takes as input a prime number and a
power-conjugate presentation for a finite soluble group, and as output produces
a full set of absolutely irreducible representations of the group over fields
whose characteristic is the specified prime, each representation being written
over its minimal field.
