The fields generated by character values of linear and unitary groups

For a finite group $G$, define the field $\mathbb{Q}(G) := \mathbb{Q}(\{χ(g) : χ\in \operatorname{Irr}(G) , g \in G\})$. In this article we compute generating sets for the fields $\mathbb{Q}(G)$ when $G$ is a finite general or special linear group, a finite general or special unitary group, or one o