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On the Existence of the Maximal Unramified Pro-$2$-Extension over the Cyclotomic $\mathbb{Z}_2$-Extension with Prescribed Metacyclic Galois Group

Mohamed Mahmoud Chems-Eddin, Hamza El Mamry

Abstract

For an integer $m\geq 2$, we aim to investigate the realizability of types of metacyclic-nonmodular groups, whose abelianization is $\mathbb{Z}/2 \mathbb{Z}\times\mathbb{Z}/2^m \mathbb{Z}$, as the Galois group of the maximal unramified $2$-extension (resp. pro-$2$-extension) over certain number fields of $2$-power degree (resp. cyclotomic $\mathbb Z_2$-extensions). Furthermore, we present some new techniques for studying Greenberg's conjecture for some number fields. In particular, the reader can find results concerning the real quadratic fields $F=\mathbb{Q}(\sqrt{ηq rs})$, the real biquadratic fields $K=\mathbb{Q}(\sqrt{ηq},\sqrt{rs})$, with $η\in\{1,2\}$, and the Fröhlich multiquadratic fields of the form $\mathbb{F}=\mathbb{Q}(\sqrt{q }, \sqrt {r}, \sqrt{s})$, where $q$, $r$ and $s$ are odd prime numbers.

On the Existence of the Maximal Unramified Pro-$2$-Extension over the Cyclotomic $\mathbb{Z}_2$-Extension with Prescribed Metacyclic Galois Group

Abstract

For an integer , we aim to investigate the realizability of types of metacyclic-nonmodular groups, whose abelianization is , as the Galois group of the maximal unramified -extension (resp. pro--extension) over certain number fields of -power degree (resp. cyclotomic -extensions). Furthermore, we present some new techniques for studying Greenberg's conjecture for some number fields. In particular, the reader can find results concerning the real quadratic fields , the real biquadratic fields , with , and the Fröhlich multiquadratic fields of the form , where , and are odd prime numbers.

Paper Structure

This paper contains 3 sections, 20 theorems, 19 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Prerequisites from groups and class field theories
  3. The main results and their proofs

Key Result

Proposition 1

Let $M$ be a metacyclic-nonmodular group whose abelianization is isomorphic to $\mathbb{Z}/2 \mathbb{Z}\times\mathbb{Z}/2^n \mathbb{Z}$, with $n >1$, and such that $M=\langle a, b\ | \ a^2\equiv b^{2^n} \equiv 1 \text{ mod } M' \rangle$. Then $M$ admits a presentation in one of the following types,

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (31)

  • Definition
  • Proposition : benjashnepreprint1993, Proposition 2
  • Theorem : Theorem \ref{['etatheoremmetacyclic-nonmodular']}
  • Corollary : Corollary \ref{['corol2']}
  • Theorem : Theorem \ref{['corllarealtri(2 2)']}
  • Theorem 2.1: aaboune, Theorem 4.2
  • Lemma 2.2: aaboune, Corollary 4.9
  • Lemma 2.3
  • proof
  • Lemma 2.4: Ku-50
  • ...and 21 more