Normal CM-fields with class number one

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.

The Gauss problem for central leaves

“…, 48} ∪ {64, 96}. The last two cases have been determined by Hofmann-Sircana [18], who showed that there is no normal CM field of degree d = 64 or d = 96 of class number one. Besides 172 abelian CM fields determined by Yamamura, there are 55 non-abelian normal CM fields with class number one under GRH.…”