Writing representations over minimal fields

Abstract

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.