On the maximal unramified pro-2-extension of $\mathbb{Z}_2$-extension of certain real biquadratic fields

TL;DR

This paper constructs explicit examples of real biquadratic fields with controlled unramified extensions, demonstrating diverse Galois groups and verifying the Greenberg conjecture in these cases.

Contribution

It provides the first known families of real biquadratic fields with specified unramified Iwasawa modules and Galois groups, advancing understanding of class field theory.

Findings

Existence of real fields with Galois group isomorphic to Q_8 or D_8

Construction of fields satisfying Greenberg conjecture

First examples of such families in literature

Abstract

For any positive integer $n$, we show that there exists a real number field $k$ (resp. $k'$) of degree $2^n$ whose $2$-class group is isomorphic $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ such that the Galois group of the maximal unramified extension of $k$ (resp. $k'$) over $k$ (resp. $k'$) is abelian (resp. non abelian, more precisely isomorphic to $Q_8$ or $D_8$, the quaternion and the dihedral group of order $8$ respectively). In fact, we construct the first examples in literature of families of real biquadratic fields whose unramified abelian Iwasawa module is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, and so that satisfying the Greenberg conjecture.