It is standard that the group of invertible matrices $GL_n(\mathbb{C})$ is dense in the space of matrices $M_n(\mathbb{C})$. We can for instance see this by the finite number of roots for the characteristic polynomial, and approach the original matrix by avoiding them.
I wonder what properties on the base field are needed for this to hold. With the above proof, we would need an infinite field with a density property to approach zero, but then it should be dense since it is a (topological) field.
Does it mean that, for a given infinite field $K$, $GL_n(K)$ is dense in $M_n(\overline{K})$ where $\overline{K}$ is something where $K$ is dense? (is there a canonical notion of "closure"? maybe we need to suppose $K$ is already given with a close field containing it?)