The class number one problem for some non-abelian normal CM-fields

Abstract

Let $\textbf {N}$ be a non-abelian normal CM-field of degree $4p,$ $p$ any odd prime. Note that the Galois group of $\textbf {N}$ is either the dicyclic group of order $4p,$ or the dihedral group of order $4p.$ We prove that the (relative) class number of a dicyclic CM-field of degree $4p$ is always greater then one. Then, we determine all the dihedral CM-fields of degree $12$ with class number one: there are exactly nine such CM-fields.