The simplest cubic fields

Shanks, Daniel

doi:10.1090/S0025-5718-1974-0352049-8

The simplest cubic fields
HTML articles powered by AMS MathViewer

by Daniel Shanks PDF

Math. Comp. 28 (1974), 1137-1152 Request permission

Abstract:

The cyclic cubic fields generated by ${x^3} = a{x^2} + (a + 3)x + 1$ are studied in detail. The regulators are relatively small and are known at once. The class numbers are always of the form ${A^2} + 3{B^2}$, are relatively large and easy to compute. The class groups are usually easy to determine since one has the theorem that if m is divisible only by ${\text {primes}} \equiv 2\pmod 3$, then the m-rank of the class group is even. Fields with different 3-ranks are treated separately.

References

H. J. Godwin and P. A. Samet, A table of real cubic fields, J. London Math. Soc. 34 (1959), 108–110. MR 100579, DOI 10.1112/jlms/s1-34.1.108
H. J. Godwin, The determination of the class-numbers of totally real cubic fields, Proc. Cambridge Philos. Soc. 57 (1961), 728–730. MR 126437, DOI 10.1017/s0305004100035854
A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
H. J. Godwin, The determination of units in totally real cubic fields, Proc. Cambridge Philos. Soc. 56 (1960), 318–321. MR 117216, DOI 10.1017/s0305004100034617
Carol Neild and Daniel Shanks, On the $3$-rank of quadratic fields and the Euler product, Math. Comp. 28 (1974), 279–291. MR 352042, DOI 10.1090/S0025-5718-1974-0352042-5
Daniel Shanks, On the conjecture of Hardy & Littlewood concerning the number of primes of the form $n^{2}+a$, Math. Comp. 14 (1960), 320–332. MR 120203, DOI 10.1090/S0025-5718-1960-0120203-6
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 10.4064/aa-21-1-71-87
B. D. Beach, H. C. Williams, and C. R. Zarnke, Some computer results on units in quadratic and cubic fields, Proceedings of the Twenty-Fifth Summer Meeting of the Canadian Mathematical Congress (Lakehead Univ., Thunder Bay, Ont., 1971) Lakehead Univ., Thunder Bay, Ont., 1971, pp. 609–648. MR 0337887

Sylow

Subgroups of Ideal Class Groups of Certain Cubic Fields

Sur les l-Classes d’Idéaux dans les Extensions Cycliques Relative de Degré Premier l

Pierre Barrucand and Harvey Cohn, A rational genus, class number divisibility, and unit theory for pure cubic fields, J. Number Theory 2 (1970), 7–21. MR 249398, DOI 10.1016/0022-314X(70)90003-X
Pierre Barrucand and Harvey Cohn, Remarks on principal factors in a relative cubic field, J. Number Theory 3 (1971), 226–239. MR 276197, DOI 10.1016/0022-314X(71)90040-0
Daniel Shanks and Larry P. Schmid, Variations on a theorem of Landau. I, Math. Comp. 20 (1966), 551–569. MR 210678, DOI 10.1090/S0025-5718-1966-0210678-1
Morris Newman, A table of the first factor for prime cyclotomic fields, Math. Comp. 24 (1970), 215–219. MR 257029, DOI 10.1090/S0025-5718-1970-0257029-5
Heinrich W. Leopoldt, Zur Geschlechtertheorie in abelschen Zahlkörpern, Math. Nachr. 9 (1953), 351–362 (German). MR 56032, DOI 10.1002/mana.19530090604
M. N. Gras, N. Moser, and J. J. Payan, Approximation algorithmique du groupe des classes de certains corps cubiques cycliques, Acta Arith. 23 (1973), 295–300 (French). MR 330099, DOI 10.4064/aa-23-3-295-300
Helmut Hasse, Arithmetische Bestimmung von Grundeinheit und Klassenzahl in zyklischen kubischen und biquadratischen Zahlkörpern, Abh. Deutsch. Akad. Wiss. Berlin. Math.-Nat. Kl. 1948 (1948), no. 2, 95 pp. (1950) (German). MR 33863

Sur le Nombre de Classes du Sous-Corps Cubique de

Marie-Nicole Gras, Méthodes et algorithmes pour le calcul numérique du nombre de classes et des unités des extensions cubiques cycliques de $\textbf {Q}$, J. Reine Angew. Math. 277 (1975), 89–116 (French). MR 389845, DOI 10.1515/crll.1975.277.89