Finite groups of Lie type as Galois groups over $\mathbb{F}_p(t)$

Abstract

In this paper a realization of all classical and most exceptional finite groups of Lie type defined over a field $\mathbb{F}_{q}$ (where $q=p^r$ is a prime power) as Galois groups over rational function fields over the prime field $\mathbb{F}_p$ is provided. Our approach runs by restricting the ground field of the groups and using criteria for bounds for Galois groups, derived from the theory of Frobenius modules.