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

Mohamed Mahmoud Chems-Eddin, Abdelkader Zekhnini, Abdelmalek Azizi

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.

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

Abstract

In this paper, we study the length of the -class field towers and the structure of the Galois groups of the maximal unramified -extensions of the layers of the cyclotomic -extension of some special Dirichlet fields. The capitulation problem is investigated too.

Paper Structure

This paper contains 4 sections, 18 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Units and $2$-class numbers of some multiquadratic fields
  3. Some Iwasawa theory results
  4. Proof of the main Theorem

Key Result

Theorem \oldthetheorem

Let $d$ be a square-free integer and $n\geq1$ a positive integer. Let $m$ be such that $2^m=h_2(-2d)$ and $K_n=\mathbb{Q}({\zeta_{2^{n+2}}, \sqrt{d} })$. Let $\mathcal{L}(K_n)$ be the maximal unramified $2$-extension of $K_{n}$ and put $G_n=\mathrm{Gal}(\mathcal{L}(K_{n})/K_{n})$. Then we have Put $G=\mathrm{Gal}(\mathcal{L}(K_{\infty})/K_{\infty})$. For the abelian case we have $G\simeq X_\inft

Figures (1)

  • Figure 1: Subfields of $\mathbb{L}/\mathbb{Q}(\sqrt 2)$

Theorems & Definitions (23)

  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem: Az-00, Lemma 5
  • Lemma \oldthetheorem: ZAT-15
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • ...and 13 more