Computation of the ideal class group of certain complex quartic fields
Author:
Richard B. Lakein
Journal:
Math. Comp. 28 (1974), 839846
MSC:
Primary 12A50
DOI:
https://doi.org/10.1090/S00255718197403740901
MathSciNet review:
0374090
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
Abstract: The ideal class group of quartic fields $K = F(\sqrt \mu )$, where $F = {\mathbf {Q}}(i)$, is calculated by a method adapted from the method of cycles of reduced ideals for real quadratic fields. The class number is found in this way for 5000 fields $K = F(\sqrt \pi ),\pi \equiv \pm 1 \bmod 4,\pi$ a prime of F. A tabulation of the distribution of class numbers shows a striking similarity to that for real quadratic fields with prime discriminant. Also, two fields were found with noncyclic ideal class group $C(3) \times C(3)$.

P. G. L. Dirichlet, "Recherches sur les formes quadratiques à coefficients et à indéterminées complexes," Werke I, pp. 533618.
 David Hilbert, Ueber den Dirichlet’schen biquadratischen Zahlkörper, Math. Ann. 45 (1894), no. 3, 309–340 (German). MR 1510866, DOI https://doi.org/10.1007/BF01446682
 A. Hurwitz, Über die Entwicklung complexer Grössen in Kettenbrüche, Acta Math. 11 (1887), no. 14, 187–200 (German). MR 1554754, DOI https://doi.org/10.1007/BF02418048
 Julius Hurwitz, Über die Reduction der Binären Quadratischen Formen mit Complexen Coefficienten und Variabeln, Acta Math. 25 (1902), no. 1, 231–290 (German). MR 1554944, DOI https://doi.org/10.1007/BF02419027 E. L. Ince, Cycles of Reduced Ideals in Quadratic Fields, British Association Tables, vol. 4, London, 1934.
 Sigekatu Kuroda, Über den Dirichletschen Körper, J. Fac. Sci. Imp. Univ. Tokyo Sect. I. 4 (1943), 383–406 (German). MR 0021031
 R. B. Lakein, A Gauss bound for a class of biquadratic fields, J. Number Theory 1 (1969), 108–112. MR 240073, DOI https://doi.org/10.1016/0022314X%2869%29900286 R. B. Lakein, "Class numbers and units of complex quartic fields," in Computers in Number Theory, Academic Press, London, 1971, pp. 167172. G. B. Mathews, "A theory of binary quadratic arithmetical forms with complex integral coefficients," Proc. London Math. Soc. (2), v. 11, 1913, pp. 329350.
 Daniel Shanks, On Gauss’s class number problems, Math. Comp. 23 (1969), 151–163. MR 262204, DOI https://doi.org/10.1090/S00255718196902622041
 Daniel Shanks and Peter Weinberger, A quadratic field of prime discriminant requiring three generators for its class group, and related theory, Acta Arith. 21 (1972), 71–87. MR 309899, DOI https://doi.org/10.4064/aa2117187
Retrieve articles in Mathematics of Computation with MSC: 12A50
Retrieve articles in all journals with MSC: 12A50
Additional Information
Article copyright:
© Copyright 1974
American Mathematical Society