Gauss’s ternary form reduction and the 2-Sylow subgroup

Shanks, Daniel

doi:10.1090/S0025-5718-1971-0297737-4

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ISSN 1088-6842(online) ISSN 0025-5718(print)

Gauss's ternary form reduction and the -Sylow subgroup

Author: Daniel Shanks
Journal: Math. Comp. 25 (1971), 837-853
MSC: Primary 12A99
Corrigendum: Math. Comp. 32 (1978), 1328-1329.
Corrigendum: Math. Comp. 32 (1978), 1328-1329.
MathSciNet review: 0297737
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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 is inaccessible. Examples are given for various d that illustrate a number of special problems.

[1] Daniel Shanks, On Gauss’s class number problems, Math. Comp. 23 (1969), 151–163. MR 0262204 (41 #6814), http://dx.doi.org/10.1090/S0025-5718-1969-0262204-1
[2] 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 (47 #4932)
[3] C. F. Gauss, Recherches Arithmétiques, Paris, 1807; reprint, Blanchard, Paris, 1953.
[4] 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 0289453 (44 #6643)
[5] Daniel Shanks, ``Solution of ${x^2} \equiv a \pmod p$ and generalizations.'' (To appear.)
[6] Daniel Shanks, New types of quadratic fields having three invariants divisible by 3, J. Number Theory 4 (1972), 537–556. MR 0313220 (47 #1775)
[7] Daniel Shanks, Solved and unsolved problems in number theory. Vol. I, Spartan Books, Washington, D.C., 1962. MR 0160741 (28 #3952)

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