Ring class fields and a result of Hasse

Abstract

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 {38,318} according as v∈M or v∉M. We then prove that v is always in the ring class field for the order Z[−27d] in F. Some of the results above are extended for subsets of Q(d) properly containing the fundamental units a+bd.