Class Numbers and Algebraic Tori

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 $\bm{Q}(ζ_p)$ and $\bm{Q}(ζ_p+ ζ_p^{-1})$ 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.

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