Geodesic
Rationality of certain algebraic tori
Izvestiya. Mathematics , Tome 5 (1971) no. 5, pp. 1049-1056

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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. 
@article{IM2_1971_5_5_a4,
     author = {V. E. Voskresenskii},
     title = {Rationality of certain algebraic tori},
     journal = {Izvestiya. Mathematics },
     pages = {1049--1056},
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     volume = {5},
     number = {5},
     year = {1971},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1971_5_5_a4/}
}
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V. E. Voskresenskii. Rationality of certain algebraic tori. Izvestiya. Mathematics , Tome 5 (1971) no. 5, pp. 1049-1056. http://geodesic.mathdoc.fr/item/IM2_1971_5_5_a4/