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 E, produces an equivalent representation su
ch that all matrix entries lie in a subfield F of E which is as small
as possible. The algorithm relies on a matrix version of Hilbert's The
orem 90, and is probabilistic with expected running time O(\E:F\d(3))
when \F\ is bounded. Using similar methods we then describe an algorit
hm which takes as input a prime number and a power-conjugate presentat
ion for a finite soluble group, and as output produces a full set of a
bsolutely irreducible representations of the group over fields whose c
haracteristic is the specified prime, each representation being writte
n over its minimal field.