Exercise 1.4.5 ($F$ is a finite field iff $\mathrm{GL}_n(F)$ is a finite group)

Proof. If $F$ is finite, then there are only finitely many matrices of type $n-n$, so ${\mathrm{GL}}_n(F)$ is a finite group .

Conversely, suppose that $F$ is infinite. Then ${\mathrm{GL}}_n(F)$ contains the matrices $\lambda I_n$, where $\lambda \in F^{\text{×}}$, therefore ${\mathrm{GL}}_n(F)$ is an infinite group.

This shows that ${\mathrm{GL}}_n(F)$ is a finite group if and only if $F$ has a finite number of elements. □