A note on the 3-rank of quadratic fields (Q1423826)

Let \(r\) and \(s\) denote the \(3\)-ranks of the ideal class groups of the imaginary quadratic field \(\mathbb Q(\sqrt d)\) and the real quadratic field \(\mathbb Q(\sqrt{-3d})\). A classical theorem of \textit{A. Scholz} [J. Reine Angew. Math. 166, 201--203 (1932; Zbl 0004.05104)] states that \(s\leq r\leq s+1\). The author gives new infinite families of such pairs of fields with \(r=s\) or \(r=s+1\), respectively.