Computation of the ideal class group of certain complex quartic fields
Author:
Richard B. Lakein
Journal:
Math. Comp. 28 (1974), 839846
MSC:
Primary 12A50
MathSciNet review:
0374090
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The ideal class group of quartic fields , where , 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 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 .
 [1]
P. G. L. Dirichlet, "Recherches sur les formes quadratiques à coefficients et à indéterminées complexes," Werke I, pp. 533618.
 [2]
D. Hilbert, "Über den Dirichletschen biquadratishen Zahlkörper," Math. Ann., v. 45, 1894, pp. 309340. (Werke I, pp. 2452) MR 1510866
 [3]
A. Hurwitz, "Über die Entwicklung komplexer Grössen in Kettenbrüche," Acta Math., v. 11, 18871888, pp. 187200. (Werke II, pp. 7283) MR 1554754
 [4]
J. Hurwitz, "Über die Reduction der binären quadratischen Formen mit complexen Coefficienten und Variabein," Acta Math., v. 25, 1902, pp. 231290. MR 1554944
 [5]
E. L. Ince, Cycles of Reduced Ideals in Quadratic Fields, British Association Tables, vol. 4, London, 1934.
 [6]
S. Kuroda, "Über den Dirichletschen Körper," J. Fac. Sci. Imp. Univ. Tokyo Sect. I, v. 4, 1943, pp. 383406. MR 9, 12. MR 0021031 (9:12f)
 [7]
R. B. Lakein, "A Gauss bound for a class of biquadratic fields," J. Number Theory, v. 1, 1969, pp. 108112. MR 39 #1427. MR 0240073 (39:1427)
 [8]
R. B. Lakein, "Class numbers and units of complex quartic fields," in Computers in Number Theory, Academic Press, London, 1971, pp. 167172.
 [9]
G. B. Mathews, "A theory of binary quadratic arithmetical forms with complex integral coefficients," Proc. London Math. Soc. (2), v. 11, 1913, pp. 329350.
 [10]
D. Shanks, "Review of table: Class number of primes of the form ," Math. Comp., v. 23, 1969, pp. 213214. MR 0262204 (41:6814)
 [11]
D. Shanks & P. Weinberger, "A quadratic field of prime discriminant requiring three generators for its class group, and related theory," Acta Arith., v. 21, 1972, pp. 7187. MR 0309899 (46:9003)
 [1]
 P. G. L. Dirichlet, "Recherches sur les formes quadratiques à coefficients et à indéterminées complexes," Werke I, pp. 533618.
 [2]
 D. Hilbert, "Über den Dirichletschen biquadratishen Zahlkörper," Math. Ann., v. 45, 1894, pp. 309340. (Werke I, pp. 2452) MR 1510866
 [3]
 A. Hurwitz, "Über die Entwicklung komplexer Grössen in Kettenbrüche," Acta Math., v. 11, 18871888, pp. 187200. (Werke II, pp. 7283) MR 1554754
 [4]
 J. Hurwitz, "Über die Reduction der binären quadratischen Formen mit complexen Coefficienten und Variabein," Acta Math., v. 25, 1902, pp. 231290. MR 1554944
 [5]
 E. L. Ince, Cycles of Reduced Ideals in Quadratic Fields, British Association Tables, vol. 4, London, 1934.
 [6]
 S. Kuroda, "Über den Dirichletschen Körper," J. Fac. Sci. Imp. Univ. Tokyo Sect. I, v. 4, 1943, pp. 383406. MR 9, 12. MR 0021031 (9:12f)
 [7]
 R. B. Lakein, "A Gauss bound for a class of biquadratic fields," J. Number Theory, v. 1, 1969, pp. 108112. MR 39 #1427. MR 0240073 (39:1427)
 [8]
 R. B. Lakein, "Class numbers and units of complex quartic fields," in Computers in Number Theory, Academic Press, London, 1971, pp. 167172.
 [9]
 G. B. Mathews, "A theory of binary quadratic arithmetical forms with complex integral coefficients," Proc. London Math. Soc. (2), v. 11, 1913, pp. 329350.
 [10]
 D. Shanks, "Review of table: Class number of primes of the form ," Math. Comp., v. 23, 1969, pp. 213214. MR 0262204 (41:6814)
 [11]
 D. Shanks & P. Weinberger, "A quadratic field of prime discriminant requiring three generators for its class group, and related theory," Acta Arith., v. 21, 1972, pp. 7187. MR 0309899 (46:9003)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
12A50
Retrieve articles in all journals
with MSC:
12A50
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197403740901
PII:
S 00255718(1974)03740901
Article copyright:
© Copyright 1974
American Mathematical Society
