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Mathematics of Computation

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



Computation of the ideal class group of certain complex quartic fields. II

Author: Richard B. Lakein
Journal: Math. Comp. 29 (1975), 137-144
MSC: Primary 12A30; Secondary 12A50
MathSciNet review: 0444605
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Abstract: For quartic fields $ K = {F_3}(\sqrt \pi )$, where $ {F_3} = Q(\rho )$ and $ \pi \equiv 1 \bmod 4$ is a prime of $ {F_3}$, the ideal class group is calculated by the same method used previously for quadratic extensions of $ {F_1} = Q(i)$, but using Hurwitz' complex continued fraction over $ Q(\rho )$. The class number was found for 10000 such fields, and the previous computation over $ {F_1}$ was extended to 10000 cases. The distribution of class numbers is the same for 10000 fields of each type: real quadratic, quadratic over $ {F_1}$, quadratic over $ {F_3}$. Many fields were found with non-cyclic class group, including the first known real quadratics with groups $ 5 \times 5$ and $ 7 \times 7$. Further properties of the continued fractions are also discussed.

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Article copyright: © Copyright 1975 American Mathematical Society