Geodesic
On two-dimensional algebraic tori. II
Izvestiya. Mathematics , Tome 1 (1967) no. 3, pp. 691-696

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It is proven that two-dimensional algebraic tori are rational over their field of definition. 
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V. E. Voskresenskii. On two-dimensional algebraic tori. II. Izvestiya. Mathematics , Tome 1 (1967) no. 3, pp. 691-696. http://geodesic.mathdoc.fr/item/IM2_1967_1_3_a10/