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Computation of the class number and class group of a complex cubic field

Authors: G. Dueck and H. C. Williams
Journal: Math. Comp. 45 (1985), 223-231
MSC: Primary 11R16; Secondary 11R29, 11Y40
Corrigendum: Math. Comp. 50 (1988), 655-657.
MathSciNet review: 790655
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Abstract: Let h and G be, respectively, the class number and the class group of a complex cubic field of discriminant $ \Delta $. A method is described which makes use of recent ideas of Lenstra and Schoof to develop fast algorithms for finding h and G. Under certain Riemann hypotheses it is shown that these algorithms will compute h in $ O(\vert\Delta {\vert^{1/5 + \varepsilon }})$ elementary operations and G in $ O(\vert\Delta {\vert^{1/4 + \varepsilon }})$ elementary operations. Finally, the results of running some computer programs to determine h and G for all pure cubic fields $ \mathcal{Q}(\sqrt[3]{D})$, with $ 2 \leqslant D < 30,000$, are summarized.

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