Geodesic
Birational equivalence of tori with a cyclic splitting field
Zapiski Nauchnykh Seminarov POMI, Rings and modules, Tome 64 (1976), pp. 153-158

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It is proved that an algebraic torus $T$ with a cyclic splitting field is birationally equivalent over the field of definition to the direct product of certain standard tori. Further, torus $T$ is stably rational over the field of definition if and only if the character modules of these standard tori are free modules. 
@article{ZNSL_1976_64_a13,
     author = {A. L. Chistov},
     title = {Birational equivalence of tori with a cyclic splitting field},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {153--158},
     publisher = {mathdoc},
     volume = {64},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1976_64_a13/}
}
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A. L. Chistov. Birational equivalence of tori with a cyclic splitting field. Zapiski Nauchnykh Seminarov POMI, Rings and modules, Tome 64 (1976), pp. 153-158. http://geodesic.mathdoc.fr/item/ZNSL_1976_64_a13/