Computation of real quadratic fields with class number one

Stephens, A. J.; Williams, H. C.

doi:10.1090/S0025-5718-1988-0958644-7

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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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