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+b√d)1/3 and v'=(a-b√d)1/3, where a+b√d 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, 3¹⁸\ 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+b√d.