A fast algorithm to compute cubic fields

Belabas, K.

doi:10.1090/S0025-5718-97-00846-6

A fast algorithm to compute cubic fields
HTML articles powered by AMS MathViewer

by K. Belabas PDF

Math. Comp. 66 (1997), 1213-1237 Request permission

Abstract:

We present a very fast algorithm to build up tables of cubic fields. Real cubic fields with discriminant up to $10^{11}$ and complex cubic fields down to $-10^{11}$ have been computed.

References

C. Batut, D. Bernardi, H. Cohen, and M. Olivier, User’s guide to pari-gp, version 1.39.03, available from ftp: megrez.math.u-bordeaux.fr, 1995.
K. Belabas, Crible et $3$-rang des corps quadratiques, Ann. de l’Inst. Fourier 46 (1996), pp. 909–949.
Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405–420. MR 491593, DOI 10.1098/rspa.1971.0075
Gilbert W. Fung and H. C. Williams, On the computation of a table of complex cubic fields with discriminant $D>-10^6$, Math. Comp. 55 (1990), no. 191, 313–325. MR 1023760, DOI 10.1090/S0025-5718-1990-1023760-X
H. Hasse, Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage, Math. Zeitschrift. 31 (1930), pp. 565–582.
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
C. Hermite, Note sur la réduction des formes homogènes à coefficients entiers et à deux indéterminées, J. reine. angew. Math. 36 (1848), pp. 357–364.
C. Hermite, Sur la réduction des formes cubiques à deux indéterminées, C.R. Acad. Sci. Paris 48 (1859), pp. 351–357.
Pascual Llorente and Jordi Quer, On totally real cubic fields with discriminant $D<10^7$, Math. Comp. 50 (1988), no. 182, 581–594. MR 929555, DOI 10.1090/S0025-5718-1988-0929555-8
G.-B. Mathews, On the reduction and classification of binary cubics which have a negative discriminant, Proc. London Math. Soc. 10 (1912), pp. 128–138.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
Takuro Shintani, On zeta-functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo Sect. I A Math. 22 (1975), 25–65. MR 0384717