Class numbers of imaginary abelian number fields

Abstract

Let $N$ be an imaginary abelian number field. We know that $h_{N}^{-}$, the relative class number of $N$, goes to infinity as $f_N$, the conductor of $N$, approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. It is known that there are only finitely many CM-fields $N$ with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of $N$. Second, we have proved in this paper that there are exactly 48 such fields.