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. 
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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/

[1] Swan R. G., “Invariant rational functions and a problem of Steenrod”, Invent. Math., 7:2 (1969), 148–158 | DOI | MR | Zbl

[2] Voskresenskii V. E., “K voprosu o stroenii podpolya invariantov tsiklicheskoi gruppy avtomorfizmov polya”, Izv. AN SSSR. Ser. matem., 34 (1970), 366–375 | MR | Zbl

[3] Masuda K., “Application of the theory of the group of classes of projective modules to the existence problem of independent parameters of invariant”, J. Math. Soc. Japan, 20:1–2 (1968), 223–232 | MR | Zbl

[4] Voskresenskii V. E., “Biratsionalnye svoistva lineinykh algebraicheskikh grupp”, Izv. AN SSSR. Ser. matem., 34 (1970), 3–19 | MR | Zbl

[5] Swan K. G., “Induced representations and projective modules”, Ann. Math., 71 (1960), 552–578 | DOI | MR | Zbl

[6] Ono T., “On the Tamagawa number of algebraic tori”, Ann. Math., 78 (1963), 47–73 | DOI | MR | Zbl