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?



Existence of Regular Semisimple elements of reductive groups in characteristic 0
A torus is unirational. The set of regular elements is open, hence also unirational. Over an infinite field, any unirational variety has (a Zariski dense set of) F-points. This answers the question in the affirmative.


