Let K = Q(ζp) and let hp be its class number. Kummer showed that p divides hp if and only if p divides the numerator of some Bernoulli number. In this expository note we discuss the generalizations of this type of criterion to totally real fields and quadratic imaginary fields.

In a recent paper we proved that there are at most finitely many complex numbers $t\ne 0,1$ such that the points $(2,\sqrt{2(2-t)})$ and $(3,\sqrt{6(3-t)})$ are both torsion on the Legendre elliptic curve defined by $y^2=x(x-1)(x-t)$. In a sequel we gave a generalization to any two points with coordinates algebraic over the field $\mathbf{Q}\left(t\right)$ and even over $\mathbf{C}\left(t\right)$. Here we reconsider the special case $(u,\sqrt{u(u-1)(u-t)})$ and $(v,\sqrt{v(v-1)(v-t)})$ with complex numbers $u$ and $v$.

Let $K=\mathbb{Q}\left(\omega \right)$, with ${\omega }^3=m$ a positive integer, be a pure cubic number field. We show that the elements $\alpha \in K^{\times }$ whose squares have the form $a-\omega$ for rational numbers $a$ form a group isomorphic to the group of rational points on the elliptic curve $E_m:y^2=x^3-m$. This result will allow us to construct unramified quadratic extensions of pure cubic number fields $K$.