Mathematics of Computation

Non-primitive number fields of degree eight and of signature

and

with small discriminant

Author(s): Schehrazad Selmane.
Journal: Math. Comp. 68 (1999), 333-344.
MSC (1991): Primary 11R11, 11R16, 11R29, 11Y40
Retrieve article in: PDF DVI PostScript
This article is available free of charge

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.

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.

Retrieve articles in Mathematics of Computation with MSC (1991): 11R11, 11R16, 11R29, 11Y40

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: 10.1090/S0025-5718-99-00998-9
PII: S 0025-5718(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
Copyright of article: Copyright 1999, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS