Ring class fields and a result of Hasse (Q6623029)

Over several years, the second author has been expounding on manifold aspects of class field theory; this includes explaining in modern terms, results from classical papers which may not be easy to read. One such aspect is the so-called Scholz reflection principle -- named after Arnold Scholz -- from 1932. In fact, Scholz reflection principle itself has more general, modern forms due to Leopoldt, Georges Gras etc. One special case of Scholz's principle states that if the class number of a real quadratic field \(\mathbb{Q}(\sqrt{d})\) is divisible by \(3\), then, so is the class number of the imaginary quadratic field \(\mathbb{Q}(\sqrt{-3d})\). Unlike quadratic fields which are characterized by their discriminants, the above principle shows that cubic fields are not. Indeed, Hasse proved in 1930 that the number of isomorphism classes of cubic fields of discriminant \(d\) is \((3^{r_3(d)}-1)/2\), where \(r_3(d)\) is the \(3\)-rank of the ideal class group of \(\mathbb{Q}(\sqrt{d})\). He proved that \(3\) divides the class number of \(\mathbb{Q}(\sqrt{-3d})\) for some square-free \(d>1\), if, and only if, there is a cubic field with the same discriminant.\N\NIn this paper, the authors prove interesting, concrete results, with a wealth of details. In particular, if the real cube root \(v = (a+b \sqrt{d})^{1/3}\) of the fundamental unit of \(\mathbb{Q}(\sqrt{d})\) belongs to the ring class field \(M\) of the order \(\mathbb{Z}[\sqrt{-3d}]\), they show that the field \(\mathbb{Q}((a+b \sqrt{d})^{1/3}+(a-b \sqrt{d})^{1/3})\) is such a cubic field. They observe that the condition \(v \in M\) is sufficient to ensure that \(3\) divides the class number of \(\mathbb{Q}(\sqrt{-3d})\) but not necessary. Further, they show that when \(v \not\in M\), the discriminant of \(\mathbb{Q}((a+b \sqrt{d})^{1/3}+(a-b \sqrt{d})^{1/3})\) is \(81~ disc(\mathbb{Q}(\sqrt{-3d}))\) or \(9~ disc(\mathbb{Q}(\sqrt{-3d}))\), according as to whether \(3|d\) or not. The proofs also yield a computation of the relative discriminant of \(\mathbb{Q}(\sqrt{-3d})(v)\) over \(\mathbb{Q}(\sqrt{-3d})\). The authors also note that the question of whether \(v\) belongs to \(M\) or not, is related to that of whether the fundamental unit \(a+b \sqrt{d}\) is a cube modulo primes expressible in the form \(x^2 +3dy^2\).\N\NThe authors also state some natural conjectures concerning a certain set of algebraic integers in \(\mathbb{Q}(\sqrt{d})\) that includes the fundamental unit; the conjectures reduce to theorems in the paper for the fundamental unit. The conjectures are backed by extensive numerical data which can be found in R. Evans, M. Van Veen, Mathematica notebook \url{https://math.ucsd.edu/~revans/CubicData.nb} 2024. The paper is a pleasure to read.