Rationality of certain algebraic tori

It is shown that the necessary condition for pure transcendence of a field of invariants given by Swan is also sufficient. We prove the rationality of factors of the type $T/G_m$, where $T$ the are tori of a certain special form, as well as the extension of the statement $H^1(k,\operatorname{Pic}\overline V)=0$ to all connected linear groups with cyclic algebraic splitting field.