Suppose $G$ is a connected reductive group defined over a field $F$ of characteristic $0$. Does every maximal torus contain a regular semisimple element defined over $F$?
I know that over an algebraically closed field this is true because being regular corresponds to being in the intersection of the complements of the kernels of the roots. Since the complement of each kernel is open and dense, we can pick an element in the intersection.
But what about if $F$ is not algebraically closed?