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Mathematics of Computation

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



Computation of real quadratic fields with class number one

Authors: A. J. Stephens and H. C. Williams
Journal: Math. Comp. 51 (1988), 809-824
MSC: Primary 11R11; Secondary 11R29, 11Y40
MathSciNet review: 958644
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Abstract: A rapid method for determining whether the real quadratic field $ \mathcal{K} = \mathcal{Q}(\sqrt D )$ has class number one is described. The method makes use of the infrastructure idea of Shanks to determine the regulator of $ \mathcal{K}$ and then uses the Generalized Riemann Hypothesis to rapidly estimate $ L(1,\chi )$ to the accuracy needed for determining whether or not the class number of $ \mathcal{K}$ is one. The results of running this algorithm on a computer for all prime values of D up to $ {10^9}$ are also presented, together with further results on runs on intervals of size $ {10^7}$ starting at $ {10^i}\,(i = 9,10, \ldots ,16)$.

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Article copyright: © Copyright 1988 American Mathematical Society

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