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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
DOI: https://doi.org/10.1090/S0025-5718-1975-0444605-4
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.


References [Enhancements On Off] (What's this?)

  • [1] A. HURWITZ, "Über die Entwicklung komplexer Grössen in Kettenbrüche," Acta Math., v. 11, 1887-1888, pp. 187-200. (Werke II, pp. 72-83). MR 1554754
  • [2] S. KURODA, "Table of class numbers $ h(p) > 1$ for quadratic fields $ Q(\sqrt p ),p \leqslant 2776817$," Math. Comp., v. 29, 1975, UMT, pp. 335-336 (this issue).
  • [3] R. B. LAKEIN, "A Gauss bound for a class of biquadratic fields," J. Number Theory, v. 1, 1969, pp. 108-112. MR 39 #1427. MR 0240073 (39:1427)
  • [4] R. B. LAKEIN, "Computation of the ideal class group of certain complex quartic fields," Math. Comp., v. 28, 1974, pp. 839-846. MR 0374090 (51:10290)
  • [5] D. SHANKS, "On Gauss's class number problems," Math. Comp., v. 23, 1969, pp. 151-163. MR 41 #6814. MR 0262204 (41:6814)
  • [6] D. SHANKS, "Review of Table: Class numbers of primes of the form $ 4n + 1$," Math. Comp., v. 23, 1969, pp. 213-214.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1975-0444605-4
Article copyright: © Copyright 1975 American Mathematical Society

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