Normal CM-fields with class number one

TL;DR

Under the assumption of the generalized Riemann hypothesis, the paper proves the non-existence of certain normal CM-fields with class number one in specific degrees by constructing comprehensive tables and applying discriminant bounds.

Contribution

The paper introduces a method to determine the non-existence of normal CM-fields with class number one in degrees 64 and 96, extending to degrees 16, 32, 56, and 82 for relative class number one, assuming GRH.

Findings

No normal CM-fields with class number one in degrees 64 and 96 assuming GRH.

Complete tables of normal CM-fields constructed using discriminant bounds.

Solved the relative class number one problem in degrees 16, 32, 56, and 82 assuming GRH.

Abstract

We show that assuming the generalized Riemann hypothesis there are no normal CM-fields with class number one of degree 64 and 96. This is done by constructing complete tables of normal CM-fields using discriminant bounds of Lee--Kwon. This solves the class number one problem for normal CM-fields assuming GRH. Using the same technique to solve the relative class number one problem in degrees 16, 32, 56 and 82, also the corresponding relative class number one problem is solved assuming GRH.