论文标题
Greenberg对真实二次字段的猜想和cyclotomic $ \ mathbb {z} _2 $ - extensions
Greenberg's conjecture for real quadratic fields and the cyclotomic $\mathbb{Z}_2$-extensions
论文作者
论文摘要
令$ \ mathcal {a} _n $为$ 2 $ - $ n $ - $ n $ - the层的$ 2 $ - $ n $ the layer $ \ mathbb {z} _2 $ extension的真实Quadratic Number Number Number Number field $ f $。 $ \ MATHCAL {A} _n $的基数与整个单元组中的Cyclotomic单元索引有关。我们提出了一种研究后一种指数的方法。作为一个应用程序，我们表明$ \ MATHCAL {a} _n $的序列稳定在实际字段中$ f = \ mathbb {q}（\ sqrt {f}）$ for integer $ 0 $ 0 <f <10000 $。格林伯格的猜想等效地适用于这些领域。
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 $0<f<10000$. Equivalently Greenberg's conjecture holds for those fields.
