Imaginary abelian number fields with class number one
Kôji Uchida · Tohoku Mathematical Journal · 1972
We have shown that there exist only a finite number of imaginary abelian number fields with relative class number h t -1 [12 or 13].There an upper bound of the conductors of such fields could be effectively determined, except for biquadratic fields of type (2, 2).Now Baker's and Stark's papers [3 and 10] show that an upper bound can be effectively determined also for those fields, because biquadratic fields of type (2, 2) with h γ = 1 are generated by imaginary quadratic fields with h γ = 1 or 2. So it is a problem of finite amount of calculation to determine all the imaginary abelian number fields with h γ = l But an upper bound we can now obtain is too large to solve this problem explicitly.In this paper, we restrict ourselves to the class number (not the relative class number) one problem, and we give some remarks and upper bounds for some cases.