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On the Narrow $2$-Class Field Tower of Some Real Quadratic Number Fields, Part I: Ranks

Elliot Benjamin, C. Snyder

TL;DR

The paper investigates the narrow 2-class field towers over real quadratic fields k with Cl_2(k) ≅ (2,2) and discriminants not expressible as sums of two squares. It develops a group-theoretic framework that links the rank of the narrow-class group Cl_2(k_+^1) to the structure of G^+/G_3^+, using genus theory, Taussky conditions, and intricate 2-group analysis to classify when the tower length is 2 versus at least 3. Through Appendices I–III, it provides exhaustive data, explicit presentations, and isomorphism-type identifications for the relevant G^+/G_3^+ and related groups, enabling precise rank determinations and laying groundwork for Part II to finalize the tower-length classification. The work highlights the deep connection between genus-theoretic decompositions, p-group structure, and class-field towers in real quadratic fields. Its results help identify which fields have finite narrow 2-class towers and guide further investigation into possible infinite towers.

Abstract

We determine some properties of the narrow 2-class field tower of those real quadratic number fields whose discriminants are not a sum of two squares and for which their 2-class groups are elementary of order $4$. Here in Part I, we determine precisely which of these fields have the 2-class groups of their narrow 2-Hilbert class fields of rank 2.

On the Narrow $2$-Class Field Tower of Some Real Quadratic Number Fields, Part I: Ranks

TL;DR

The paper investigates the narrow 2-class field towers over real quadratic fields k with Cl_2(k) ≅ (2,2) and discriminants not expressible as sums of two squares. It develops a group-theoretic framework that links the rank of the narrow-class group Cl_2(k_+^1) to the structure of G^+/G_3^+, using genus theory, Taussky conditions, and intricate 2-group analysis to classify when the tower length is 2 versus at least 3. Through Appendices I–III, it provides exhaustive data, explicit presentations, and isomorphism-type identifications for the relevant G^+/G_3^+ and related groups, enabling precise rank determinations and laying groundwork for Part II to finalize the tower-length classification. The work highlights the deep connection between genus-theoretic decompositions, p-group structure, and class-field towers in real quadratic fields. Its results help identify which fields have finite narrow 2-class towers and guide further investigation into possible infinite towers.

Abstract

We determine some properties of the narrow 2-class field tower of those real quadratic number fields whose discriminants are not a sum of two squares and for which their 2-class groups are elementary of order . Here in Part I, we determine precisely which of these fields have the 2-class groups of their narrow 2-Hilbert class fields of rank 2.
Paper Structure (10 sections, 6 theorems, 35 equations, 3 tables)

This paper contains 10 sections, 6 theorems, 35 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Real Quadratic Number Fields with ${2}$-Class Groups of Rank ${2}$ and whose Discriminants are Not a Sum of Two Squares
  3. Appendix I $d_k=d_1d_2d_3d_4$
  4. Review of the Structure of ${\operatorname{Gal}(k^2/k)}$ when ${\operatorname{Cl}_2(k)\simeq (2,2)}$
  5. Appendix II
  6. Some Properties of Relevant ${2}$-Groups
  7. The Structure of ${G^+/G_3^+}$ for ${G^+=\operatorname{Gal}(k_+^2/k)}$
  8. Appendix III
  9. The Rank of $\operatorname{Cl}_2(k_+^1)$
  10. On the Length of the Narrow $2$-Class Field Tower of ${k}$

Key Result

Lemma 1

Let $G$ be a finite $2$-group such that $G/G_2\simeq (2,2,2)$ and $G_2/G_3\simeq (2,2)$, and hence as given in one of the rows of Table 1 above. Then (a) $c_{ji\ell}\equiv c_{ij\ell}^{-1} \bmod G"$, for all $i,j,\ell\in \{1,2,3\}$; (b) $c_{123}\equiv c_{132} \bmod G_4$.

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • ...and 1 more