Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On counting totally imaginary number fields

A. Raghuram, Qiyao Yu

Abstract

A number field is said to be a CM-number field if it is a totally imaginary quadratic extension of a totally real number field. We define a totally imaginary number field to be of CM-type if it contains a CM-subfield, and of TR-type if it does not contain a CM-subfield. For quartic totally imaginary number fields when ordered by discriminant, we show that about 69.95% are of TR-type and about 33.05% are of CM-type. For a sextic totally imaginary number field we classify its type in terms of its Galois group and possibly some additional information about the location of complex conjugation in the Galois group.

On counting totally imaginary number fields

Abstract

A number field is said to be a CM-number field if it is a totally imaginary quadratic extension of a totally real number field. We define a totally imaginary number field to be of CM-type if it contains a CM-subfield, and of TR-type if it does not contain a CM-subfield. For quartic totally imaginary number fields when ordered by discriminant, we show that about 69.95% are of TR-type and about 33.05% are of CM-type. For a sextic totally imaginary number field we classify its type in terms of its Galois group and possibly some additional information about the location of complex conjugation in the Galois group.
Paper Structure (24 sections, 5 theorems, 22 equations, 4 figures)

This paper contains 24 sections, 5 theorems, 22 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Examples
  5. Quartic extensions
  6. Sextic totally imaginary extensions
  7. $6$-transitive groups
  8. A digression on possible Galois groups
  9. The main result on sextic totally imaginary number fields
  10. Proof of Theorem \ref{['thm:sextic']}
  11. Preliminary remarks
  12. $G = C_6$
  13. $G = S_3$
  14. $G = S_5$
  15. $G = S_6$
  16. ...and 9 more sections

Key Result

Theorem 3.1

When ordered by absolute discriminant, asymptotically, about $66.948\%$ of the totally imaginary quartic fields are of TR-type, and the remaining $33.052\%$ of the totally imaginary quartic fields are of CM-type, i.e., $\lim_{X \to \infty} R_4^{\textbf{CM}}(X) \approx 0.33052.$

Figures (4)

  • Figure 1: Subgroup Lattice of $D_4$
  • Figure 2: Subgroup Lattice of $D_6$
  • Figure 3: Subgroup Lattice of $S_4$ (subgroups of $S_3$ omitted)
  • Figure :

Theorems & Definitions (7)

  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • Proposition 4.1