Quadratic Fields with Special Class Groups

Abstract

For every prime number $p \geq 5$ it is shown that, under certain hypotheses on $x \in {\mathbf {Q}}$, the imaginary quadratic fields ${\mathbf {Q}}(\sqrt {{x^{2p}} - 6{x^p} + 1} )$ have ideal class groups with noncyclic p-parts. Several numerical examples with $p = 5$ and 7 are presented. These include the field \[ {\mathbf {Q}}(\sqrt { - 4805446123032518648268510536} ).\] The 7-part of its class group is isomorphic to $C(7) \times C(7) \times C(7)$, where $C(n)$ denotes a cyclic group of order n.