Ring class fields and a result of Hasse

For squarefree d>1, let M denote the ring class field for the order Z-3d in F=Q (-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+bd) ^1/3 and v'= (a-bd) ^1/3, where a+bd is the fundamental unit in Q (d). We prove that E can be taken as Q (v+v') if and only if v M. As byproducts of the proof, we give explicit congruences for a and b which hold if and only if v M, and we also show that the norm of the relative discriminant of F (v) /F lies in \1, 3⁶\ or \3⁸, 3^{18\} according as v M or v M. Some of the results above are extended for subsets of Q (d) properly containing the fundamental units a+bd.