Finitely Generated General Linear Group Is there a $F$ over which $GL_n(F)$, the general linear group, is finitely generated? Obviously one generator won't work, since $GL_n(F)$ is never abelian for any field. This is possibly a silly question, which I can't really trace to any single source (just sort of came to me one day). By the way, $n$ neither has to be a fixed number nor arbitrary, so interpret the question as you wish (just so long as your interpretation doesn't render the question trivial). EDIT: As Morgan Rodgers points out, the question becomes relatively trivial by taking $F$ to be finite. So, perhaps I should stipulate that $F$ is infinite. What's the answer in that case?

If $\text{GL}_m(F)$ is finitely generated, then $F^*$ is finitely generated (take determinants of a finite set of generators). Note that any subgroup of a finitely generated Abelian group is finitely generated. If $F$ has characteristic zero, $F^*$ contains a copy of $\Bbb Q^*$ which is not finitely generated.

If $F$ has characteristic $p$, and infinite, then either it is (i) algebraic over $\Bbb F_p$ or (ii) transcendental. In (i) the group $F^*$ is torsion, so cannot be finitely generated (lest it be finite). In (ii) $F^*$ contains a copy of the $\Bbb F_p(X)^*$ which is not finitely generated.