论文标题
Weber的班级问题和$ p $ - 合理性中的cyclotomic $ \ wideHat {\ mathbb {z}} $ - $ \ mathbb {q} $的扩展
Weber's class number problem and $p$-rationality in the cyclotomic $\widehat{\mathbb{Z}}$-extension of $\mathbb{Q}$
论文作者
论文摘要
令$ k $为$ n $ th lay plocyclomic $ \ wideHat {\ mathbb {z}} $ - 扩展$ \ wideHat {\ mathbb {q}} $。许多作者（Aoki，Fukuda，Horie，Ichimura，Inatomi，Komatsu，Komatsu，Miller，Morisawa，Nakajima，Nakajima，Okazaki，Washington，\，$ \ ldots $）分析了$ P $ -P $ -CLASS组$ {\ MATHCAL C} _K $。我们以更概念性的形式重新访问此问题，因为计算表明$ p $ torsion $ {\ Mathcal t} _k $的最大Abelian $ p $ P $ P $ -RAMIFIED PRAMIFIED PRAMIFIED PRAMIFIED PRAMIFIED PRAMIFIED PRAMIFIED PRAMIFIED PRAMIFIED PRAMIFIED PRAMIFIED PRAMIFIED PRAMIFIED PRAMIFIEN-P $ - $ k $（tate-shafarevich of $ k $ k $ k $ k $）通常是非trivial;这引起了问题，因为$ \＃{\ mathcal t} _k = \＃{\ Mathcal C} _k \ \，\＃{\ Mathcal r} _k $其中$ {\ Mathcal r} _k $ as is p $ p $ addic监管机构。我们给出了一种新方法测试$ {\ Mathcal t} _k \ ne 1 $（定理4.6，表6.2），并表征$ p $ -extenSions $ f $ of $ k $ in $ \ wideHat {\ mathbb {q}} $，带有$ {\ mathcal c} _F \ ne 1 $ 7.6.6和theorem（theorem）。我们发布易于使用的程序，再次证明八个已知示例，并允许进一步的大量计算。
Let $K$ be the $N$th layer in the cyclotomic $\widehat{\mathbb{Z}}$-extension $\widehat{\mathbb{Q}}$. Many authors (Aoki, Fukuda, Horie, Ichimura, Inatomi, Komatsu, Miller, Morisawa, Nakajima, Okazaki, Washington,\,$\ldots$) analyse the behavior of the $p$-class groups ${\mathcal C}_K$. We revisit this problem, in a more conceptual form, since computations show that the $p$-torsion group ${\mathcal T}_K$ of the Galois groups of the maximal abelian $p$-ramified pro-$p$-extension of $K$ (Tate--Shafarevich group of $K$) is often non-trivial; this raises questions since $\# {\mathcal T}_K = \# {\mathcal C}_K\, \# {\mathcal R}_K$ where ${\mathcal R}_K$ is the normalized $p$-adic regulator. We give a new method testing ${\mathcal T}_K \ne 1$ (Theorem 4.6, Table 6.2) and characterize the $p$-extensions $F$ of $K$ in $\widehat{\mathbb{Q}}$ with ${\mathcal C}_F \ne 1$ (Theorem 7.5 and Corollary 7.6). We publish easy to use programs, justifying again the eight known examples, and allowing further extensive computations.
