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

Elliot Benjamin, Mohamed Mahmoud Chems-Eddin

TL;DR

This work investigates the narrow $2$-class field towers of real quadratic fields $\ k$ whose $2$-class group is elementary of order $4$ and whose discriminants are not sums of two squares, focusing on fields with $\mathrm{G}/\mathrm{G}_3 \simeq 64.150$ (types $(c_3)$ and $(d_1)$). The authors establish heuristic criteria linking tower length $t$ to the $8$-rank of the narrow $2$-class group $\mathrm{Cl}_2(\mathrm{k}^1_+)$, and employ the Kuroda class-number formula for $V_4$-extensions to distinguish $t=2$ from $t\ge 3$, introducing unramified quadratic extensions $\mathrm{L}_j$ and unit-indices $q$ as central tools. They provide explicit constructions of the unit groups $E_{\mathrm{k}^1}$ and $E_{\mathrm{k}^1_+}$ across several discriminant configurations (cases $(c_3)$ and $(d_1)$), using Wada's method and norm computations to obtain concrete generators; these detailed unit descriptions underpin the class-number computations and the proposed conjectures that, in this 64.150 family, the narrow $2$-class tower length is always $t=2$. The paper also offers computed examples and a framework for extending these results to broader families via Ambiguous Class Number Formula calculations and explicit unit analyses.

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.

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

TL;DR

This work investigates the narrow -class field towers of real quadratic fields whose -class group is elementary of order and whose discriminants are not sums of two squares, focusing on fields with (types and ). The authors establish heuristic criteria linking tower length to the -rank of the narrow -class group , and employ the Kuroda class-number formula for -extensions to distinguish from , introducing unramified quadratic extensions and unit-indices as central tools. They provide explicit constructions of the unit groups and across several discriminant configurations (cases and ), using Wada's method and norm computations to obtain concrete generators; these detailed unit descriptions underpin the class-number computations and the proposed conjectures that, in this 64.150 family, the narrow -class tower length is always . The paper also offers computed examples and a framework for extending these results to broader families via Ambiguous Class Number Formula calculations and explicit unit analyses.

Abstract

In this article we continue the investigation of the length of the narrow -class field tower of real quadratic number fields whose discriminants are not a sum of two squares and for which their -class groups are elementary of order . Letting equal the Galois group of the second Hilbert narrow -class field over , and denote the lower central series of , we give heuristic evidence that the length of the narrow -class field tower of is equal to when is of type (in the tables of Hall and Senior). We also give the formulation of the relevant unit groups of the narrow Hilbert -class field for these fields.
Paper Structure (9 sections, 22 theorems, 35 equations, 3 tables)

This paper contains 9 sections, 22 theorems, 35 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Narrow $2$-class field tower: Results and Examples
  3. Prerequisites to Obtain the Unit Groups of $\mathrm{k}^1$ and $\mathrm{k}_+^{1}$
  4. Investigation of Units of $\mathrm{k}^1$ and $\mathrm{k}_+^{1}$
  5. Units of $\mathrm{k}^1$ and $\mathrm{k}_+^{1}$ when $d$ Satisfies the Conditions $(c_3)$ and $p_i\equiv 3\pmod 4$, for $i=1$, $2$, $3$, $4$
  6. Units of $\mathrm{k}^1$ and $\mathrm{k}_+^{1}$ when $d$ Satisfies the Conditions $(c_3)$ and $d_4=-8$
  7. Units of $\mathrm{k}^1$ and $\mathrm{k}_+^{1}$ when $d$ Satisfies the Conditions $(c_3)$ and $d_i=-8$, for $i=1$, $2$ or $3$
  8. Units of $\mathrm{k}^1$ and $\mathrm{k}_+^{1}$ when $d$ Satisfies the Conditions $(d_1)$
  9. Appendix

Key Result

Lemma 2.1

If $\mathbf{C}l_2(\mathrm{k}^1_+)$ has $8$-$\mathrm{rank} =0$, then $\mathrm{k}$ has narrow $2$-class field tower length $t=2$.

Theorems & Definitions (32)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.4: Snyder
  • Lemma 2.5
  • proof
  • Lemma 3.1: azizunints99, Proposition 3
  • Lemma 3.2: azizunints99, Proposition 2
  • Lemma 3.3: Ku-50
  • Lemma 3.4: Az-00, Lemme 5
  • Lemma 3.5: BenSnyder25PartII, Lemma 6
  • ...and 22 more