Construction of Number Fields of Odd Degree with Class Numbers Divisible by Three, Five or by Seven

Abstract

We introduce a simple way to construct a family of number fields of given degree with class numbers divisible by a given integer, by using the arithmetic theory of elliptic curves. In particular, we start with an elliptic curve defined over the rational number field with a rational torsion point of order l ∈ {3,5,7}, and show a way to construct infinitely many number fields of given odd degree d ≥ 3 with class numbers divisible by l.