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Extensions with Galois group Hol(C8) unramified over a complex quadratic number field

Elliot Benjamin, Franz Lemmermeyer, Chip Snyder

TL;DR

This work classifies unramified extensions of complex quadratic fields with Galois group $Hol(C_8)$, showing the Galois group over a quadratic subfield must be either the semi-dihedral group $SD_{16}$ or the modular group $M_4(2)$. Through a combination of group-theoretic analysis, capitulation theory, and classical class-field theory, the authors describe when the 2-class field tower can realize these groups and provide explicit constructions. They derive a concrete description of ${Gal}(k^2/\mathbb{Q})$ and give constructive procedures for generating unramified ${SD}_{16}$- and ${M}_4(2)$-extensions of complex quadratic fields, relying on genus fields, Gaussian integers, and norm equations. The results connect transfer kernels, capitulation gaps, and Scholz reciprocity to yield explicit, verifiable families of fields with prescribed 2-class towers, with computations corroborated by GAP.

Abstract

We study normal extensions with Galois group Hol($C_8$) that are unramified over a complex quadratic subfield. The Galois group is either the semi-dihedral group or the modular group of order $16$. We present an explicit construction of such fields.

Extensions with Galois group Hol(C8) unramified over a complex quadratic number field

TL;DR

This work classifies unramified extensions of complex quadratic fields with Galois group , showing the Galois group over a quadratic subfield must be either the semi-dihedral group or the modular group . Through a combination of group-theoretic analysis, capitulation theory, and classical class-field theory, the authors describe when the 2-class field tower can realize these groups and provide explicit constructions. They derive a concrete description of and give constructive procedures for generating unramified - and -extensions of complex quadratic fields, relying on genus fields, Gaussian integers, and norm equations. The results connect transfer kernels, capitulation gaps, and Scholz reciprocity to yield explicit, verifiable families of fields with prescribed 2-class towers, with computations corroborated by GAP.

Abstract

We study normal extensions with Galois group Hol() that are unramified over a complex quadratic subfield. The Galois group is either the semi-dihedral group or the modular group of order . We present an explicit construction of such fields.

Paper Structure

This paper contains 5 sections, 24 theorems, 43 equations, 6 figures, 2 tables.

Table of Contents

  1. Group Theoretical Calculations
  2. Capitulation Gaps
  3. A few results from class field theory
  4. Unramified ${\operatorname{SD}}_{16}$-extensions of complex quadratic number fields
  5. Unramified ${\operatorname{M}}_4(2)$-extensions of complex quadratic number fields

Key Result

Proposition 1

Let $F$ be a number field with odd class number, $L/F$ a normal $2$-extension with Galois group $G$, $H$ a subgroup of index $2$, and let $k = L^H$ denote the fixed field of $H$. If $L/k$ is unramified, then $G$ is generated by elements of order $2$ lying outside of $H$.

Figures (6)

  • Figure 1: Subgroups of ${\operatorname{SD}}_{16}$
  • Figure 2: Subextensions of the $2$-class field tower of ${\mathbb Q}(\sqrt{-pq}\,)$ (with two conjugate fields omitted)
  • Figure 3: $2$-class groups of the subextensions of $L/k$ for unramified ${\operatorname{SD}}_{16}$-extensions of $k = {\mathbb Q}(\sqrt{-pq}\,)$
  • Figure 4: Subgroups of ${\operatorname{M}}_4(2)$
  • Figure 5: The extension $F'/{\mathbb Q}(i,\sqrt{pq}\,)$.
  • ...and 1 more figures

Theorems & Definitions (33)

  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Theorem 5
  • proof
  • Corollary 6
  • proof
  • Corollary 7
  • Theorem 8
  • ...and 23 more