Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

The computation of sextic fields with a cubic subfield and no quadratic subfield


Author: M. Olivier
Journal: Math. Comp. 58 (1992), 419-432
MSC: Primary 11R21; Secondary 11R32, 11Y40
DOI: https://doi.org/10.1090/S0025-5718-1992-1106977-7
MathSciNet review: 1106977
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We describe six tables of sixth-degree fields K containing a cubic subfield k and no quadratic subfield: one for totally real sextic fields, one for sextic fields with four real places, two for sextic fields with two real places, and two for totally imaginary sextic fields (depending on whether the cubic subfield is totally real or not). The tables provide for each possible discriminant $ {d_K}$ of K a quadratic polynomial which defines K/k , the discriminant of the cubic subfield and the Galois group of a Galois closure $ N/\mathbb{Q}$ of $ K/\mathbb{Q}$.


References [Enhancements On Off] (What's this?)

  • [1] I. O. Angell, A table of complex cubic fields, Bull. London Math. Soc. 5 (1973), 37-38. MR 0318099 (47:6648)
  • [2] -, A table of totally real cubic fields, Math. Comp. 30 (1976), 184-187. MR 0401701 (53:5528)
  • [3] A.-M. Bergé, J. Martinet, and M. Olivier, The computation of sextic fields with a quadratic subfield, Math. Comp. 54 (1990), 869-884. MR 1011438 (90k:11169)
  • [4] G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), 863-911. MR 695893 (84f:20005)
  • [5] V. Ennola and R. Turunen, On totally real cubic fields, Math. Comp. 44 (1985), 495-518. MR 777281 (86e:11100)
  • [6] U. Fincke and M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comp. 44 (1985), 463-471. MR 777278 (86e:11050)
  • [7] H. J. Godwin, The determination of fields of small discriminant with a given subfield, Math. Scand. 6 (1958), 40-46. MR 0105404 (21:4146)
  • [8] H. J. Godwin and P. Samet, A table of real cubic fields, J. London Math. Soc. 34 (1959), 108-110. MR 0100579 (20:7009)
  • [9] P. Llorente and E. Nart, Effective determination of the decomposition of the rational primes in a cubic field, Proc. Amer. Math. Soc. 87 (1983), 579-585. MR 687621 (84d:12003)
  • [10] P. Llorente and J. Quer, On totally real cubic fields with discriminant $ D < {10^7}$, Math. Comp. 50 (1988), 581-594. MR 929555 (89g:11099)
  • [11] J. Martinet, Méthodes géométriques dans la recherche des petits discriminants, Progr. Math., vol. 59, Birkhäuser, Boston, 1985, pp. 147-179. MR 902831 (88h:11083)
  • [12] -, Discriminants and permutation groups, Number Theory (Richard A. Mollin, ed.), Walter de Gruyter, Berlin and New York, 1990, pp. 359-385. MR 1106673 (92c:11126)
  • [13] M. Olivier, Corps sextiques primitifs, Ann. Inst. Fourier (Grenoble) 40 (1990), 757-767. MR 1096589 (92a:11123)
  • [14] M. Pohst, On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields, J. Number Theory 14 (1982), 99-117. MR 644904 (83g:12009)
  • [15] D. Shanks, Review of I. O. Angell, "Table of complex cubic fields", Math. Comp. 29 (1975), Review 33, 661-665.
  • [16] P. Smadja, Calculs effectifs sur les idéaux des corps de nombres algébriques, Univ. d'Aix-Marseille, U.E.R. de Luminy, 1976.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11R21, 11R32, 11Y40

Retrieve articles in all journals with MSC: 11R21, 11R32, 11Y40


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1992-1106977-7
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society