Divisibility of class numbers of imaginary quadratic fields whose discriminant has only two prime factors

Abstract

Let $g \geq 2$ and $n \geq 1$ be integers. In this paper, we shall show that there are infinitely many imaginary quadratic fields whose class number is divisible by $2g$ and whose discriminant has only two prime divisors. As a corollary, we shall show that there are infinitely many imaginary quadratic fields whose 2-class group is a cyclic group of order divisible by $2^{n}$.