Ring class fields and a result of Hasse

Abstract

For squarefree $d>1$, let M denote the ring class field for the order $Z\left[\sqrt{-3d}\right]$ in $F=Q\left(\sqrt{-3d}\right)$. 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 $v={(a+b\sqrt{d})}^{1/3}$ and $v^{\prime }={(a-b\sqrt{d})}^{1/3}$, where $a+b\sqrt{d}$ is the fundamental unit in $Q\left(\sqrt{d}\right)$. We prove that E can be taken as $Q(v+v^{\prime })$ if and only if $v\in M$. As byproducts of the proof, we give explicit congruences for a and b which hold if and only if $v\in M$, and we also show that the norm of the relative discriminant of $F\left(v\right)/F$ lies in $\{1,3^6\}$ or $\{3^8,3^{18}\}$ according as $v\in M$ or $v\notin M$. We then prove that v is always in the ring class field for the order $Z\left[\sqrt{-27d}\right]$ in F. Some of the results above are extended for subsets of $Q\left(\sqrt{d}\right)$ properly containing the fundamental units $a+b\sqrt{d}$.