Ring class fields and a result of Hasse

For squarefree d > 1, let M denote the ring class field for the order Z[root-3d] in F = Q(root-3d). Hasse proved that 3 divides the class number of F if and only if there exists a cubic extension E of Q such that E and F have the same discriminant. Define the real cube roots v = (a + b root d)(1/3) and v ' = (a - b root d)(1/3), where a + b root d is the fundamental unit in Q(root d). We prove that E can be taken as Q(v + v ') if and only if v is an element of M. As byproducts of the proof, we give explicit congruences for a and b which hold if and only if v is an element of M, and we also show that the norm of the relative discriminant of F(v)/F lies in {1, 3(6)} or {3(8), 3(18)} according as v is an element of M or v is not an element of M. We then prove that v is always in the ring class field for the order Z[root-27d] in F. Some of the results above are extended for subsets of Q(root d) properly containing the fundamental units a + b root d. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.