I saw this in a paper that I've been reading and I've been trying to figure out if this is true or not.
Let $G(F)$ be a affine, simple, connected, adjoint, algebraic group over a local field endowed with the topology induced by the local field. Suppose that $H$ is a Zariski-dense subgroup of $G(F)$ that is open with respect to this topology. Is $H$ necessarily finite index in $G$?
Can someone provide a hint?
Edit: The local field is non-archimedean.