Imaginary Non-CM Fields of $2$-Power Degree
Abstract
We establish a new method for the study of imaginary non-CM fields of arbitrary 2-power degree and class number one. In particular, we show that for degrees greater than 16, dihedral fields either lie in the ray class field of one of the three imaginary quadratic fields ℚ(√(−2)), ℚ(√(−3)), ℚ(√(−67)), with the conductor of this ray class field supported only on primes dividing 2 and with explicit upper bounds on the relevant exponents, or are Hilbert class fields of imaginary quadratic fields. We also give a complete classification of the 104 imaginary non-CM fields of degree 16 with class number one, providing explicit tables of defining polynomials, Galois groups, base fields, and conductors. These 104 non-CM fields, together with the five CM fields classified by Louboutin and Okazaki \cite{lou7}, complete the classification of all imaginary fields of degree 16 with class number one. For CM fields, the problem reduces to computing relative class numbers via analytic class number formulae, whereas for non-CM fields the absence of a totally real subfield of index 2 necessitates a further approach and leads to a richer variety of Galois groups and ramification patterns.
Related
- Maximal unramified 3-extensions of imaginary quadratic fields and \textrm SL\sb 2(\mathbb Z\sb 3)
- Bowen--Franks groups and minus class groups of imaginary abelian fields
- An infinite family of imaginary biquadratic fields with a large class number
- p-class groups in the cyclotomic ℤp-extension of imaginary quadratic fields
- Automorphic symbols and automorphic \emph{L}-values of GL(2) over imaginary quadratic fields
BibTeX
@misc{amiri2026imaginary,
title = {{Imaginary Non-CM Fields of $2$-Power Degree}},
author = {Farahnaz Amiri},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24114},
eprint = {2609.24114},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24114},
}
RIS
TY - UNPB
TI - Imaginary Non-CM Fields of $2$-Power Degree
AU - Farahnaz Amiri
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.24114
UR - https://arxiv.org/abs/2609.24114
AB - We establish a new method for the study of imaginary non-CM fields of arbitrary 2-power degree and class number one. In particular, we show that for degrees greater than 16, dihedral fields either lie in the ray class field of one of the three imaginary quadratic fields ℚ(√(−2)), ℚ(√(−3)), ℚ(√(−67)), with the conductor of this ray class field supported only on primes dividing 2 and with explicit upper bounds on the relevant exponents, or are Hilbert class fields of imaginary quadratic fields. We also give a complete classification of the 104 imaginary non-CM fields of degree 16 with class number one, providing explicit tables of defining polynomials, Galois groups, base fields, and conductors. These 104 non-CM fields, together with the five CM fields classified by Louboutin and Okazaki \cite{lou7}, complete the classification of all imaginary fields of degree 16 with class number one. For CM fields, the problem reduces to computing relative class numbers via analytic class number formulae, whereas for non-CM fields the absence of a totally real subfield of index 2 necessitates a further approach and leads to a richer variety of Galois groups and ramification patterns.
ER -