On birational equivalence of algebraic tori (Q844430)

The author studies the birational classification of algebraic tori over non-closed fields. Given a field \(k\), let \({\overline k}\) be the algebraic closure of \(k\). A \(k\)-torus is an algebraic group \(T\) such that \(T\otimes_k{\overline k}\) is isomorphic to a connected diagonal group over \({\overline k}\). Two \(k\) varieties \(X_1\) and \(X_2\) are said to be stably equivalent if there exist \(p,m\in{\mathbb N}\) such that \(X_1\times {\mathbb A}^p\) is birationally isomorphic to \(X_2\times {\mathbb A}^m\). A variety is stably rational if it is stably equivalent to affine space. The author proves the conjecture that if a torus is rationally stable, then it is rational.