On bases of groups of circular units of some imaginary abelian number fields (Q1352652)

For any abelian number field \(K\), the class number \(h^+\) of its maximal real subfield is a factor of the index of the group of circular units \(C\) in the unit group \(E\), i.e. \[ (E : C) = c h^+. \] This was known to E. Kummer for \(K=\mathbb Q (\zeta_{p^k})^+\) (and \(c=1\)), and generally shown by \textit{W. Sinnott} [Invent. Math. 62, 181--234 (1980; Zbl 0465.12001)]. If one has a basis for the group \(C\) at one's disposal, matters simplify considerably and the cofactor \(c\) can be given explicitly. This was done for the case \(K=\mathbb Q(\zeta_n)\) by \textit{R. Gold} and \textit{J. Kim} [Compos. Math. 71, 13--27 (1989; Zbl 0687.12003)] and by \textit{R. Kučera} [J. Number Theory 40, 284--316 (1992; Zbl 0744.11052)]. In the present paper this task is solved for imaginary fields \(K\) with conductor divisible by at most two different primes, supposing that the maximal subfields of \(K\) with conductor a prime power are also imaginary.