Relevance of $\mathrm{GL}(n, \mathbb{Q})$

Question

When studying topics such as representation theory, Lie theory or even just linear algebra, we see the vector spaces $\mathrm{GL}(n, \mathbb{K})$ for $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$ all the time. General linear groups over finite fields also have their applications and the general linear group $\mathrm{GL}(2, \mathbb{Q}_p)$ is immensely important in local Langlands Correspondence. Therefore, I asked myself whether there are any mathematical applications, e.g. in the aforementioned fields of study, for $\mathrm{GL}(n, \mathbb{Q})$ or in general for $\mathrm{GL}(n, \mathbb{K})$ where $\mathbb{K}$ is of characteristic $0$, but not algebraically closed.