Fields generated by characters of finite linear groups

In previous work, the authors confirmed the speculation of J. G. Thompson that certain multiquadratic fields are generated by specified character values of sufficiently large alternating groups $A_n$. Here we address the natural generalization of this speculation to the finite general linear groups ${\mathrm{G}\mathrm{L}}_m\left({\mathbb{F}}_q\right)$ and ${\mathrm{S}\mathrm{L}}_2\left({\mathbb{F}}_q\right)$.