On the class numbers of certain number fields obtained from points on elliptic curves II

Abstract

We construct a family of cyclic extensions of number fields, in which every finite place is unramified, from an elliptic curve with a rational torsion point. As an application, we obtain such polynomials $F(X)$ of rational coefficients that have the following property: For a rational number $\xi$ chosen at random, the class number of the field generated by the square root of $F(\xi)$ is ``often'' divisible by 3, 5 or by 7.