Rational invariants of certain classical groups over finite fields

Abstract

Let \( \mathbb{F}_q \) be a finite field with q elements, where q is a prime power. Let G be a subgroup of the general linear group over \( \mathbb{F}_q \) and \( \mathbb{F}_q \) be the rational function field over \( \mathbb{F}_q \). We seek to understand the structure of the rational invariant subfield \( \mathbb{F}_q \). In this paper, we prove that \( \mathbb{F}_q \) is rational (or, purely transcendental) by giving an explicit set of generators when G is the symplectic group. In particular, the set of generators we gave satisfies the Dickson property.