Infrastructure of ambiguous ideal classes of orders of real quadratic fields (Q1190628)

The authors intend to study ``Shanks infrastructure'' in the orders of a real quadratic field [cf. \textit{D. Shanks}, Proc. 1972 Number Theory Conf. Boulder, Colorado, 217-224 (1972; Zbl 0334.12005)]. Let \({\mathcal O}_{\mathcal D}\) be an order of a real quadratic field with discriminant \({\mathcal D}\). Then, they first define the ``symmetric ideal'', which is necessarily reduced, associated with certain decomposition of \({\mathcal D}\) into the sum of two squares, and prove the following: Let \({\mathcal C}\) be a primitive ambiguous non-principal class of order \({\mathcal O}_{\mathcal D}\) which contains exactly \(\ell\) primitive reduced ideals. Then, in the case of \(N(\varepsilon_{\mathcal D})=-1\), \(\ell\) is odd and \({\mathcal C}\) contains a reduced ambiguous ideal and a symmetric ideal. In the case of \(N(\varepsilon_{\mathcal D})=+1\), \(\ell\) is even and \({\mathcal C}\) contains either two reduced ambiguous ideals or two symmetric ideals. Moreover, they show that the symmetric ideal \({\mathcal S}'\) constructed with the symmetric ideal \({\mathcal S}\) in an ambiguous class in certain simple manner is always principal or equivalent to \({\mathcal S}\) according to \(N(\varepsilon_{\mathcal D})=-1\) or \(+1\).