On the ideal class group of real biquadratic fields

Abstract:

We discuss the structure of the ideal class group of real biquadratic fields $K$, concentrating on the case that the $4$-rank of the ideal class groups of the quadratic subfields of $K$ is $0$. In this case, we give estimates for the $4$-rank of the ideal class group of $K$. As an example, let $K = \mathbb {Q}(\sqrt p ,\sqrt {627} )$, where $p$ is a prime satisfying certain congruence conditions. The $2$-primary part of the ideal class group of $K$ is then isomorphic to ${(\mathbb {Z}/4\mathbb {Z})^2},\mathbb {Z}/4\mathbb {Z} \times {(\mathbb {Z}/2\mathbb {Z})^2}$, or ${(\mathbb {Z}/2\mathbb {Z})^4}$. Further, each of the above occurs infinitely often.