Siegel-Ramachandra invariants generate ray class fields (Q1941355)

Let \(K\) be an imaginary quadratic field and \({\mathcal{O}}_K\) its ring of integers. For an ideal \(\mathfrak{f} \not = (1)\) of \(K\), let denote \(Cl(\mathfrak{f})\) the ray class group of \(K\) modulo \(\mathfrak{f}\) and \(K_{\mathfrak{f}}\) the ray class field of \(K\) modulo \(\mathfrak{f}\). For \(C \in Cl(\mathfrak{f})\) let \(g_{\mathfrak{f}}(C)\) denote the Siegel-Ramachandra invariant. Then it is known that \(g_{\mathfrak{f}}(C)\) is an algebraic integer in \(K_{\mathfrak{f}}\) and \(g_{\mathfrak{f}}(C)^{\sigma(C')}=g_{\mathfrak{f}}(CC')\) for \(C' \in Cl(\mathfrak{f})\), where \(\sigma\) : \(Cl(\mathfrak{f}) \rightarrow\text{Gal}(K_{\mathfrak{f}}/K)\) denotes the Artin map. Starting from this result, the author proves the following: Let \(\mathfrak{f}=\Pi_{i=1}^{r}\mathfrak{p}_i^{n_i}\) satisfy that the qoutient group \(({\mathcal{O}}_K/\mathfrak{p}_i^{n_i})^{\times}/ \{ \alpha \mod \mathfrak{p}_i^{n_i} \in ({\mathcal{O}}_K/\mathfrak{p}_i^{n_i})^{\times} \; | \; \alpha \in \mathcal{O}_K^{\times}\}\) has exponet \(>2\) for evry \(i\). Then for any \(C \in Cl(\mathfrak{f})\) and any nonzero integer \(n\), we have \(K_{\mathfrak{f}}=K(g_{\mathfrak{f}}(C)^n)\). Using the invariants of this type, \textit{K. Ramachandra} [Ann. Math. (2) 80, 104--148 (1964; Zbl 0142.29804)] first studied the generators, systems of units, and the class numbers of the ray class fields. His results, however, are rather complicate. So several authors (for instance, \textit{R. Schertz} [J. Théor. Nombres Bordx. 9, No. 2, 383--394 (1997; Zbl 0902.11047)]) have studied about more simple constructions of generators of the ray class field \(K_{\mathfrak{f}}\) under certain conditions using \(g_{\mathfrak{f}}(C)\). The result of the paper is along with this direction.