On counting totally imaginary number fields
Number Theory
2024-01-31 v1
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.
Keywords
Cite
@article{arxiv.2401.16586,
title = {On counting totally imaginary number fields},
author = {A. Raghuram and Qiyao Yu},
journal= {arXiv preprint arXiv:2401.16586},
year = {2024}
}
Comments
16 pages including an appendix with 3 figures