Explicit realization of the Dickson groups \(G_2(q)\) as Galois groups. (Q1880103)

We already know that the Dickson group \(G_2(q)\) occurs as a Galois group of a regular Galois extension of regular function fields over \({\mathbb{F}}_q\). Using a recent result of \textit{B. H. Matzat} [Galois theory and modular forms, Dev. Math. 11, 233--268 (2004; Zbl 1111.12002)], the author gives an explicit polynomial \(f\in{\mathbb{F}}_q(t,u)[X]\) having Galois group \(G_2(q)\) over \({\mathbb{F}}_q(t,u)\).