New number fields with known $p$-class tower

Abstract

The $p$-class tower $ F^{\infty}_{p} (k)$ of a number field $k$ is its maximal unramified pro-$p$ extension. It is considered to be known when the $p$-tower group, that is the Galois group
$G := F^{\infty}_{p} (k)|k)$, can be identified by an explicit presentation. The main intention of this article is to characterize assigned nite 3-groups uniquely by abelian quotient invariants of subgroups of nite index, and to provide evidence of actual realizations of these groups by 3-tower groups $G$ of real quadratic fields $K = \mathbb{Q}(\sqrt{d}) $ with 3-capitulation type (0122) or (2034).