Gauss’s ternary form reduction and the $2$-Sylow subgroup
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- Math. Comp. 25 (1971), 837-853 Request permission
Corrigendum: Math. Comp. 32 (1978), 1328-1329.
Corrigendum: Math. Comp. 32 (1978), 1328-1329.
Abstract:
An algorithm is developed for determining the 2-Sylow subgroup of the class group of a quadratic field provided the complete factorization of the discriminant d is known. It uses Gauss’s ternary form reduction with some new improvements and is applicable even if d is so large that the class number $h(d)$ is inaccessible. Examples are given for various d that illustrate a number of special problems.References
- Daniel Shanks, On Gauss’s class number problems, Math. Comp. 23 (1969), 151–163. MR 262204, DOI 10.1090/S0025-5718-1969-0262204-1
- Daniel Shanks, Class number, a theory of factorization, and genera, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 415–440. MR 0316385 C. F. Gauss, Recherches Arithmétiques, Paris, 1807; reprint, Blanchard, Paris, 1953.
- 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 Daniel Shanks, “Solution of ${x^2} \equiv a \pmod p$ and generalizations.” (To appear.)
- Daniel Shanks, New types of quadratic fields having three invariants divisible by $3$, J. Number Theory 4 (1972), 537–556. MR 313220, DOI 10.1016/0022-314X(72)90027-3
- Daniel Shanks, Solved and unsolved problems in number theory. Vol. I, Spartan Books, Washington, D.C., 1962. MR 0160741
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Math. Comp. 25 (1971), 837-853
- MSC: Primary 12A99
- DOI: https://doi.org/10.1090/S0025-5718-1971-0297737-4
- MathSciNet review: 0297737