A finiteness theorem for imaginary abelian number fields

Lately, I. Miyada proved that there are only finitely many imaginary abelian number fields with Galois groups of exponents ≤2 with one class in each genus. He also proved that under the assumption of the Riemann hypothesis there are exactly 301 such number fields. Here, we prove the following finiteness theorem: there are only finitely many imaginary abelian number fields with one class in each genus. We note that our proof would make it possible to find an explict upper bound on the discriminants of these number fields which are neither quadratic nor biquadratic bicyclic. However, we do not go into any explicit determination.