Greenberg's conjecture for real quadratic fields and the cyclotomic Z2-extensions

Abstract

Let $\mathcal{A}_n$ be the $2$-part of the ideal class group of the $n$-th layer of the cyclotomic $\mathbb{Z}_2$-extension of a real quadratic number field $F$. The cardinality of $\mathcal{A}_n$ is related to the index of cyclotomic units in the full group of units. We present a method to study the latter index. As an application we show that the sequence of the $\mathcal{A}_n$'s stabilizes for the real fields $F=\mathbb{Q}(\sqrt{f})$ for any integer $0

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实二次场的Greenberg猜想及分环z2扩展

设$\mathcal{A}_n$是实二次元域$F$的环切$\mathbb{Z}_2$扩展的$n$第n$层理想类群的$2$部分。$\mathcal{A}_n$的基数与整组环切单位的索引有关。本文提出了一种研究后一指标的方法。作为一个应用，我们证明了$\mathcal{A}_n$的序列对于实域$F=\mathbb{Q}(\sqrt{F})$对于任意整数$0< F <10000$是稳定的。格林伯格的猜想同样适用于这些领域。

本文章由计算机程序翻译，如有差异，请以英文原文为准。