For some research work, I need to know the classification of elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$, up to conjugation.
Since I essentially need $n\le 4$, I think that I can show it by hand, using cyclotomic extensions and Galois theory, but is there some work in the literature on this?
EDIT: Looking at the possible orders is essentially trivial in $\mathrm{GL}(n,\mathbb{Q})$, by just looking at the cyclotomic polynomials. The conjugacy classes require a little more work but are easy exercises, at least in low dimension. For $\mathrm{PGL}(n,\mathbb{Q})$, the case of orders prime to $n$ follows essentially from the case of $\mathrm{GL}(n,\mathbb{Q})$, the orders are more interesting.