Voskresenskii, Valentin E. Stably rational algebraic tori. (English) Zbl 0946.14030 J. Théor. Nombres Bordx. 11, No. 1, 263-268 (1999). Suppose that \(T\) is a torus over a field \(k\) of characteristic zero. If \(T\) is stably rational, that is, \(T\times_k{\mathbb A}^m_k\) is rational over \(k\) for some \(m\), then it is conjectured that \(T\) is rational over \(k\). Here this conjecture is proved under the additional assumption that the splitting field of \(T\) is a cyclic extension of \(k\), by reformulating it in terms of linear representations of \(T\). The method is first used to reprove a weaker result of A. A. Klyachko [in: Arithmetic and geometry of varieties, Interuniv. Collect. Sci. Works, Kujbyshev, 73-78 (1988; Zbl 0751.14031)] and then extended to give the main result of the paper. Reviewer: G.K.Sankaran (Bath) Cited in 1 Document MSC: 14M20 Rational and unirational varieties 14G25 Global ground fields in algebraic geometry 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14L99 Algebraic groups Keywords:stably rational torus; cyclic splitting field; rationality of torus Citations:Zbl 0751.14031 × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML EMIS References: [1] Beauville, A., Colliot-Thélène, J.-L., Sansuc, J.-J., Swinnerton-Dyer, Sir P., Variétés stablement rationnelles non rationnelles. Ann. Math.121 (1985), 283-318. · Zbl 0589.14042 [2] Voskresenski, V., The geometry of linear algebraic groups. Proc. Steklov Inst. Math.132 (1973), 173-183. · Zbl 0309.14040 [3] Voskresenski, V., Fields of Invariants of Abelian Groups. Russian Math. Surveys28 (1973), 79-105. · Zbl 0289.14006 [4] Klyachko, A., On the rationality of tori with a cyclic splitting field. Arithmetic and Geometry of Varieties, Kuibyshev Univ., 1988, 73-78 (Russian). · Zbl 0751.14031 [5] Chistov, A., On the birational equivalence of tori with a cyclic splitting field. Zapiski Nauchnykh Seminarov LOMI64 (1976), 153-158 (Russian). · Zbl 0358.14017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.