WRITING REPRESENTATIONS OVER MINIMAL FIELDS

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.