Cyclic-sixteen class fields for $\textbf {Q}((-p)^{1/2})$ by modular arithmetic
HTML articles powered by AMS MathViewer
- by Harvey Cohn PDF
- Math. Comp. 33 (1979), 1307-1316 Request permission
Abstract:
Numerical experiments result in the construction of cyclic-sixteen class fields for ${\mathbf {Q}}{( - p)^{1/2}}$, p prime $< 2000$, by radicals involving quadratic and biquadratic parameters. These fields are characterized by rational factorization properties modulo a variable prime; but it suffices to use only three primes selected and checked by computer to verify the class field, if earlier work (jointly with Cooke) on the cyclic-eight class field is utilized.References
- Pierre Barrucand and Harvey Cohn, Note on primes of type $x^{2}+32y^{2}$, class number, and residuacity, J. Reine Angew. Math. 238 (1969), 67–70. MR 249396, DOI 10.1515/crll.1969.238.67
- Helmut Bauer, Zur Berechnung der $2$-Klassenzahl der quadratischen Zahlkörper mit genau zwei verschiedenen Diskriminantenprimteilern, J. Reine Angew. Math. 248 (1971), 42–46 (German). MR 289453, DOI 10.1515/crll.1971.248.42
- Harvey Cohn and George Cooke, Parametric form of an eight class field, Acta Arith. 30 (1976), no. 4, 367–377. MR 422209, DOI 10.4064/aa-30-4-367-377 P. G. L. DIRICHLET, "Untersuchungen über die Theorie der complexen Zahlen," J. Reine Angew. Math., v. 22, 1841, pp. 375-378.
- Helmut Hasse, Kurt Hensels entscheidender Anstoss zur Entdeckung des Lokal-Global-Prinzips, J. Reine Angew. Math. 209 (1962), 3–4 (German). MR 139510, DOI 10.1515/crll.1962.209.3 E. L. INCE, Cycles of Reduced Ideals in Quadratic Fields, British Assoc. Adv. Sci. Math. Tables, vol. IV, London, 1934. R. LAKEIN, Class Number and Fundamental Unit of Dirichlet Fields With Prime Relative Discriminant. (Unpublished table.) K. S. WILLIAMS, "On the divisibility of the class number of ${\mathbf {Q}}{( - p)^{1/2}}$ by 16." (Manuscript.)
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp. 33 (1979), 1307-1316
- MSC: Primary 16A05
- DOI: https://doi.org/10.1090/S0025-5718-1979-0537976-5
- MathSciNet review: 537976