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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Quadratic fields with special class groups

Author: James J. Solderitsch
Journal: Math. Comp. 59 (1992), 633-638
MSC: Primary 11R29; Secondary 11R11, 11Y40
MathSciNet review: 1139091
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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

$\displaystyle {\mathbf{Q}}(\sqrt { - 4805446123032518648268510536} ).$

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.

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Article copyright: © Copyright 1992 American Mathematical Society