Could we prove the surjectivity of Lang's map for $GL(N)$ without using algebraic geometry?
In other words, given a invertible matrix $M$ in $GL_N(\mathbb{F})$, there exists another invertible matrix $P$ in $GL_N(\mathbb{F})$ such that $P^{-1}P^{(q)}=M$.
Here $\mathbb{F}$ denotes as an algebraically closed field $\mathbb{F}_p$. And $P^{(q)}$ is the coordinate-wise $q$-power where $q$ is a fixed power of $p$.