On the Hilbert $2$-class field towers of some cyclotomic $\mathbb{Z}_2$-extensions

TL;DR

This paper investigates the structure and length of 2-class field towers in certain cyclotomic $bZ_2$-extensions, analyzing Galois groups and capitulation phenomena in unramified 2-extensions of specific Dirichlet fields.

Contribution

It provides new insights into the Galois group structures and tower lengths of 2-class fields in cyclotomic $bZ_2$-extensions of special Dirichlet fields.

Findings

Determined the length of 2-class field towers for specific cyclotomic extensions.

Analyzed the structure of Galois groups of maximal unramified 2-extensions.

Investigated capitulation phenomena in these extensions.

Abstract

In this paper, we study the length of the $2$-class field towers and the structure of the Galois groups $\mathrm{Gal}(\mathcal{L}(K_n)/K_n)$ of the maximal unramified $2$-extensions of the layers $K_n$ of the cyclotomic $\mathbb{Z}_2$-extension of some special Dirichlet fields. The capitulation problem is investigated too.