On the Narrow 2-Class Field Tower of Some Real Quadratic Number Fields: Lengths Heuristics Follow-Up

TL;DR

This paper investigates the length of narrow 2-class field towers in certain real quadratic fields, providing heuristic evidence that the tower length is typically 2 under specific Galois group conditions.

Contribution

It offers new heuristic evidence linking Galois group structures to the tower length in real quadratic fields with elementary 2-class groups.

Findings

Heuristic evidence that tower length is 2 when G/G_3 is of type 64.150.

Formulation of the relevant unit groups of the narrow Hilbert 2-class field.

Identification of conditions under which the tower length is exactly 2.

Abstract

In this article we continue the investigation of the length of the narrow $2$-class field tower of real quadratic number fields $\mathrm{k}$ whose discriminants are not a sum of two squares and for which their $2$-class groups are elementary of order $4$. Letting $\mathrm{G}$ equal the Galois group of the second Hilbert narrow $2$-class field over $\mathrm{k}$, and $[\mathrm{G}_i]$ denote the lower central series of $\mathrm{G}$, we give heuristic evidence that the length of the narrow $2$-class field tower of $\mathrm{k}$ is equal to $2$ when $\mathrm{G}/\mathrm{G}_3$ is of type $64.150$ (in the tables of Hall and Senior). We also give the formulation of the relevant unit groups of the narrow Hilbert $2$-class field for these fields.