Stably rational algebraic tori
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 \textit{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.
scientific article; zbMATH DE number 1370116