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Infrastructure of ambiguous ideal classes of orders of real quadratic fields. (Infrastructure des classes ambiges d’idéaux des ordres des corps quadratiques réels.) (French) Zbl 0756.11030

The authors intend to study “Shanks infrastructure” in the orders of a real quadratic field [cf. 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 \(+1\).
Reviewer: H.Yokoi (Nagoya)

MSC:

11R11 Quadratic extensions
11R29 Class numbers, class groups, discriminants
11R54 Other algebras and orders, and their zeta and \(L\)-functions

Citations:

Zbl 0334.12005
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