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Non-primitive number fields of degree eight
and of signature $(2,3)$, $(4,2)$ and $(6,1)$
with small discriminant


Author: Schehrazad Selmane
Journal: Math. Comp. 68 (1999), 333-344
MSC (1991): Primary 11R11, 11R16, 11R29, 11Y40
DOI: https://doi.org/10.1090/S0025-5718-99-00998-9
MathSciNet review: 1489974
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Abstract: We give the lists of all non-primitive number fields of degree eight having two, four and six real places of discriminant less than 6688609, 24363884 and 92810082, respectively, in absolute value. For each field in the lists, we give its discriminant, the discriminant of its subfields, a relative polynomial generating the field over one of its subfields and its discriminant, the corresponding polynomial over $\mathbf Q$, and the Galois group of its Galois closure.


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

  • 1. H. Anai, M. Noro and K. Yokoyama, Computation of the splitting fields and the Galois groups of polynomials, Algorithms in Algebraic Geometry and Applications (Santander, 1994), Progr. Math., vol. 143, Birkhäuser, Basel, 1996, pp. 29-50. MR 98a:12003
  • 2. Ch. Batut, D. Bernadi, H. Cohen and M. Olivier, GP/PARI Calculator version 1.37, Publ. Université Bordeaux 1 (1991).
  • 3. G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Alg. 11 (1983), 863-911. MR 84f:20005
  • 4. H. Cohen and F. Diaz y Diaz, A polynomial reduction algorithm, Séminaire de Théorie des Nombres Bordeaux 3 (1991), 351-360. MR 93a:11107
  • 5. F. Diaz y Diaz, Tables minorant la racine $n$-ième du discriminant d'un corps de degré $n$, Publ. Math. Orsay 80.06 (1980). MR 82i:12007
  • 6. F. Diaz y Diaz, Private communication to the author.
  • 7. F. Diaz y Diaz, Petits discriminants des corps de nombres totalement imaginaries de degré 8, J. Number Theory. 25 (1987), 34-52. MR 88a:11115
  • 8. F. Diaz y Diaz and M. Oliver, Corps imprimitifs de degré $9$ de petit discriminant, Preprint.
  • 9. H. J. Godwin, On quartic fields of signature one with small discriminant, Quart. J. Math. Oxford Ser. (2) 8 (1957), 214-222. MR 20:3844
  • 10. H. J. Godwin, Real quartic fields with small discriminant, J. London Math. Soc. 31 (1956), 478-485. MR 18:565b
  • 11. A. Leutbecher, Euclidean fields having a large Lenstra constant, Ann. Inst. Fourier, Grenoble 35, 2 (1985), 83-106. MR 86j:11107
  • 12. J. Martinet, Méthodes géométriques dans la recherche des petits discriminants, Sém. de Théorie des nombres de Paris 1983/84, Birkhäuser Verlag, Basel (1985), 147-179.
  • 13. J. Martinet, Petits discriminants des corps de nombres, ``Journées Arithmétiques 1980'' (J. V. Armitage, ed.), London Math. Soc. Lecture Note Series, vol. 56, Cambridge Univ. Press, 1982, pp. 151-193. MR 84g:12009
  • 14. M. Noro and T. Takeshima, Risa/Asir-a computer algebra system, Proc. ISSAC 92, ACM Press, 1992, pp. 387-396.
  • 15. M. Pohst, On the computation of number fields of small discriminant including the minimum discriminants of sixth degree fields, J. Number Theory 14 (1982), 99-117. MR 83g:12009
  • 16. M. Pohst, J. Martinet and F. Diaz y Diaz, The minimum discriminant of totally real octic fields, J. Number Theory 36 (1990), 145-159. MR 91g:11128
  • 17. M. Pohst, On computing isomorphisms of equation orders, Math. Comp. 48 (1987), 309-314. MR 88b:11066
  • 18. G. Poitou, Sur les petits discriminant, Séminaire Delange-Pisot-Poitou (Théorie des Nombres) 18-ième année, 1976/1977, Exposé 6, 1-18. MR 81i:12007
  • 19. A. Valibouze, Théorie de Galois constructive, Mémoire d'Habilitation à Diriger les Recherches, Université Pierre et Marie Curie, 1994.

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Additional Information

Schehrazad Selmane
Affiliation: Université des Sciences et de la Technologie Houari Boumediene, Institut de Mathematiques, B.P. 32, El-Alia, Bab-Ezzouar 16111, Algiers, Algeria
Email: selmane@ist.cerist.dz

DOI: https://doi.org/10.1090/S0025-5718-99-00998-9
Keywords: Quadratic fields, quartic fields, relative extensions, discriminant.
Received by editor(s): March 1, 1995
Received by editor(s) in revised form: September 11, 1996
Article copyright: © Copyright 1999 American Mathematical Society

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