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

TL;DR

This paper investigates the realizability of certain metacyclic Galois groups as the Galois group of maximal unramified 2-extensions over specific number fields, and introduces new techniques related to Greenberg's conjecture.

Contribution

It presents new methods for studying the existence of particular metacyclic Galois groups over number fields and explores their realization in unramified 2-extensions.

Findings

Results on realizability over real quadratic fields.

Findings on biquadratic and multiquadratic fields.

New techniques for Greenberg's conjecture.

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 $F=\mathbb{Q}(\sqrt{\eta q rs})$, the real biquadratic fields $K=\mathbb{Q}(\sqrt{\eta q},\sqrt{rs})$, with $\eta\in\{1,2\}$, and the Fr\"ohlich multiquadratic fields of the form $\mathbb{F}=\mathbb{Q}(\sqrt{q }, \sqrt {r}, \sqrt{s})$, where $q$, $r$ and $s$ are odd prime numbers.