二面体扩张的范数一环面的合理性问题
Rationality problem for norm one tori for dihedral extensions
AI总结：
本文利用Endo与Miyata及Endo的定理，完整解决了Galois闭包为二面体扩张的范数一环面的合理性（稳定等价）问题，并在附录中细化了稳定有理情形的证明。
摘要：
我们借助Endo和Miyata [EM75, 定理1.5, 定理2.3]以及Endo [End11, 定理2.1]的结果，对Galois闭包L/k为二面体扩张的K/k的范数一环面R^{(1)}_{K/k}(\mathbb{G}_m)的合理性问题（直至稳定k-等价）给出了完整解答。通过类似技术，我们在论文附录中细化了Endo和Miyata定理中稳定有理情形的证明。
英文摘要：
We give a complete answer to the rationality problem (up to stable $k$-equivalence) for norm one tori $R^{(1)}_{K/k}(\mathbb{G}_m)$ of $K/k$ whose Galois closures $L/k$ are dihedral extensions with the aid of Endo and Miyata [EM75, Theorem 1.5, Theorem 2.3] and Endo [End11, Theorem 2.1]. By using a similar technique, we give refinements of the proof of stably rational cases of Endo and Miyata's theorems as an appendix of the paper.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
