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

Lakein, Richard B.

doi:10.1090/S0025-5718-1975-0444605-4

Computation of the ideal class group of certain complex quartic fields. II
HTML articles powered by AMS MathViewer

by Richard B. Lakein PDF

Math. Comp. 29 (1975), 137-144 Request permission

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

A. Hurwitz, Über die Entwicklung complexer Grössen in Kettenbrüche, Acta Math. 11 (1887), no. 1-4, 187–200 (German). MR 1554754, DOI 10.1007/BF02418048

Math. Comp.

R. B. Lakein, A Gauss bound for a class of biquadratic fields, J. Number Theory 1 (1969), 108–112. MR 240073, DOI 10.1016/0022-314X(69)90028-6
Richard B. Lakein, Computation of the ideal class group of certain complex quartic fields, Math. Comp. 28 (1974), 839–846. MR 374090, DOI 10.1090/S0025-5718-1974-0374090-1
Daniel Shanks, On Gauss’s class number problems, Math. Comp. 23 (1969), 151–163. MR 262204, DOI 10.1090/S0025-5718-1969-0262204-1

Math. Comp.