Imprimitive ninth-degree number fields with small discriminants

Diaz y Diaz, F.; Olivier, M.

doi:10.1090/S0025-5718-1995-1260128-X

Imprimitive ninth-degree number fields with small discriminants
HTML articles powered by AMS MathViewer

by F. Diaz y Diaz and M. Olivier PDF

Math. Comp. 64 (1995), 305-321 Request permission

Abstract:

We present tables of ninth-degree nonprimitive (i.e., containing a cubic subfield) number fields. Each table corresponds to one signature, except for fields with signature (3,3), for which we give two different tables depending on the signature of the cubic subfield. Details related to the computation of the tables are given, as well as information about the CPU time used, the number of polynomials that we deal with, etc. For each field in the tables, we give its discriminant, the discriminant of its cubic subfields, the relative polynomial generating the field over one of its cubic subfields, the corresponding (irreducible) polynomial over $\mathbb {Q}$, and the Galois group of the Galois closure. Fields having interesting properties are studied in more detail, especially those associated with sextic number fields having a class group divisible by 3.

References

I. O. Angell, A table of complex cubic fields, Bull. London Math. Soc. 5 (1973), 37–38. MR 318099, DOI 10.1112/blms/5.1.37
I. O. Angell, A table of totally real cubic fields, Math. Comput. 30 (1976), no. 133, 184–187. MR 0401701, DOI 10.1090/S0025-5718-1976-0401701-6
A.-M. Bergé, J. Martinet, and M. Olivier, The computation of sextic fields with a quadratic subfield, Math. Comp. 54 (1990), no. 190, 869–884. MR 1011438, DOI 10.1090/S0025-5718-1990-1011438-8

User’s guide to PARI-GP version

Johannes Buchmann and David Ford, On the computation of totally real quartic fields of small discriminant, Math. Comp. 52 (1989), no. 185, 161–174. MR 946599, DOI 10.1090/S0025-5718-1989-0946599-1
Johannes Buchmann, David Ford, and Michael Pohst, Enumeration of quartic fields of small discriminant, Math. Comp. 61 (1993), no. 204, 873–879. MR 1176706, DOI 10.1090/S0025-5718-1993-1176706-0
Gregory Butler and John McKay, The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), no. 8, 863–911. MR 695893, DOI 10.1080/00927878308822884
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
F. Diaz y Diaz, Petits discriminants des corps de nombres totalement imaginaires de degré $8$, J. Number Theory 25 (1987), no. 1, 34–52 (French, with English summary). MR 871167, DOI 10.1016/0022-314X(87)90014-X
F. Diaz y Diaz, A table of totally real quintic number fields, Math. Comp. 56 (1991), no. 194, 801–808, S1–S12. MR 1068820, DOI 10.1090/S0025-5718-1991-1068820-3
F. Diaz y Diaz, A table of totally real quintic number fields, Math. Comp. 56 (1991), no. 194, 801–808, S1–S12. MR 1068820, DOI 10.1090/S0025-5718-1991-1068820-3
Veikko Ennola and Reino Turunen, On totally real cubic fields, Math. Comp. 44 (1985), no. 170, 495–518. MR 777281, DOI 10.1090/S0025-5718-1985-0777281-8

On the computation of the maximal order in a Dedekind domain

David Ford and Michael Pohst, The totally real $A_5$ extension of degree $6$ with minimum discriminant, Experiment. Math. 1 (1992), no. 3, 231–235. MR 1203877, DOI 10.1080/10586458.1992.10504261
Hiroyuki Fujita, The minimum discriminant of totally real algebraic number fields of degree $9$ with cubic subfields, Math. Comp. 60 (1993), no. 202, 801–810. MR 1176709, DOI 10.1090/S0025-5718-1993-1176709-6
Kurt Girstmair, On invariant polynomials and their application in field theory, Math. Comp. 48 (1987), no. 178, 781–797. MR 878706, DOI 10.1090/S0025-5718-1987-0878706-1

KANT—a tool for computations in algebraic number fields

Armin Leutbecher, Euclidean fields having a large Lenstra constant, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 2, 83–106 (English, with French summary). MR 786536, DOI 10.5802/aif.1011
P. Llorente and A. V. Oneto, On the real cubic fields, Math. Comp. 39 (1982), no. 160, 689–692. MR 669661, DOI 10.1090/S0025-5718-1982-0669661-3
Jacques Martinet, Methodes géométriques dans la recherche des petits discriminants, Séminaire de théorie des nombres, Paris 1983–84, Progr. Math., vol. 59, Birkhäuser Boston, Boston, MA, 1985, pp. 147–179 (French). MR 902831
M. Olivier, Corps sextiques contenant un corps quadratique. I, Sém. Théor. Nombres Bordeaux (2) 1 (1989), no. 1, 205–250 (French). MR 1050276, DOI 10.5802/jtnb.15
Michel Olivier, Corps sextiques contenant un corps quadratique. II, Sém. Théor. Nombres Bordeaux (2) 2 (1990), no. 1, 49–102 (French). MR 1061760, DOI 10.5802/jtnb.20
Michel Olivier, Corps sextiques contenant un corps cubique. III, Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 201–245 (French). MR 1116107, DOI 10.5802/jtnb.48
Michel Olivier, Corps sextiques primitifs. IV, Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 2, 381–404 (French). MR 1149805, DOI 10.5802/jtnb.58

Computation of Galois groups for polynomials with degree up to eleven

Michael Pohst, On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields, J. Number Theory 14 (1982), no. 1, 99–117. MR 644904, DOI 10.1016/0022-314X(82)90061-0
M. Pohst, J. Martinet, and F. Diaz y Diaz, The minimum discriminant of totally real octic fields, J. Number Theory 36 (1990), no. 2, 145–159. MR 1072461, DOI 10.1016/0022-314X(90)90069-4
M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory, Encyclopedia of Mathematics and its Applications, vol. 30, Cambridge University Press, Cambridge, 1989. MR 1033013, DOI 10.1017/CBO9780511661952
A. Schwarz, M. Pohst, and F. Diaz y Diaz, A table of quintic number fields, Math. Comp. 63 (1994), no. 207, 361–376. MR 1219705, DOI 10.1090/S0025-5718-1994-1219705-3
Richard P. Stauduhar, The determination of Galois groups, Math. Comp. 27 (1973), 981–996. MR 327712, DOI 10.1090/S0025-5718-1973-0327712-4