Class Numbers and Algebraic Tori

Abstract

Let $p$ be an odd prime number, $D_p$ be the dihedral group of order $2p$, $h_p$ and $h_p^{+}$ be the class numbers of $\mathbf{Q}({\zeta }_p)$ and $\mathbf{Q}({\zeta }_p+{\zeta }_p^{-1})$ respectively. Theorem. $h_p^{+}=1$ if and only if, for any field $k$ admitting a $D_p$-extension, all the algebraic $D_p$-tori over $k$ are stably rational. A similar result for $h_p=1$ and $C_p$-tori is valid also.

A 2-page appendix is added

References in corpus (2)

• Rationality problem for algebraic tori
• Algebraic tori - thirty years after