The relative class number of certain imaginary abelian fields (Q912915)

Let \(m\neq 1\) be a square-free integer, prime to an odd prime \(p\), and let \(K=\mathbb Q(\sqrt{m},\zeta)\) where \(\zeta\) denotes a primitive \(p^ n\)-th root of unity (\(n\) a positive integer). The author considers the minus parts \(\mathbb Z[G]^-\), and \(S^-\) of the group ring \(\mathbb Z[G]\) \((G=\text{Gal}(K/\mathbb Q))\), and the Stickelberger ideal \(S\) of \(\mathbb Z[G]\). He mentions a special system of generators of \(S^-\) considered as a \(\mathbb Z\)-module, and he expresses the group index \((\mathbb Z[G]^- : S^-)\) by means of a product of two special determinants. Using a formula of \textit{K. Iimura} [Arch. Math. 36, 45--52 (1981; Zbl 0465.12002)] he obtains a new formula for the relative class number \(h^-_ K\) of \(K\).