Elements of order four in the narrow class group of real quadratic fields (Q2788667)

Let \(k\) be a real quadratic number field with discriminant \(d_k\). It is well known that the \(2\)-torsion \(\mathrm{Cl}^+(k)[2]\) of the narrow class group of \(k\) is generated by the narrow ideal classes containing the ramified prime ideals of \(k\). The analogous result for the ideal class group fails to be true if and only if \(d_k\) can be written as a sum of two squares and the fundamental unit \(\varepsilon_k\) of \(k\) has positive norm. In [\textit{F. Lemmermeyer}, J. Aust. Math. Soc. 93, No. 1--2, 115--120 (2012; Zbl 1294.11190)] a complete set of generators is obtained for \(\mathrm{Cl}(k)[2]\) under the assumption that the discriminant is odd and is a sum of two squares. The paper under review extends this result to the case of even discriminant. As in the odd case, the described generators come from the decompositions of \(d_k\) as sums of squares. The proof is achieved by exploiting a correspondence between narrow ideal classes and cycles of reduced quadratic forms. Motivated by two explicit examples, the authors conclude their paper with a section concerning a characterization of when two of their generators are equivalent in the narrow sense.