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Tables of octic fields with a quartic subfield. (English) Zbl 1036.11066

Summary: We describe the computation of extended tables of degree 8 fields with a quartic subfield, using class field theory. In particular we find the minimum discriminants for all signatures and for all the possible Galois groups. We also discuss some phenomena and statistics discovered while making the tables, such as the occurrence of 11 non-isomorphic number fields having the same discriminant, or several pairs of non-isomorphic number fields having the same Dedekind zeta function.
The method of computation given in the earlier paper [H. Cohen, F. Diaz y Diaz and M. Olivier, ibid. 67, 773–795 (1998; Zbl 0929.11064)] is used.

MSC:

11Y40 Algebraic number theory computations
11R37 Class field theory
11R21 Other number fields

Citations:

Zbl 0929.11064
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Full Text: DOI

References:

[1] K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), no. 219, 1213 – 1237. · Zbl 0882.11070
[2] Gregory Butler and John McKay, The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), no. 8, 863 – 911. · Zbl 0518.20003 · doi:10.1080/00927878308822884
[3] Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. · Zbl 0786.11071
[4] Henri Cohen and Francisco Diaz y Diaz, A polynomial reduction algorithm, Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 2, 351 – 360 (English, with French summary). · Zbl 0758.11053
[5] H. Cohen, F. Diaz y Diaz, and M. Olivier, Computing ray class groups, conductors and discriminants, Math. Comp. 67 (1998), no. 222, 773 – 795. · Zbl 0929.11064
[6] J. Conway, A. Hulpke and J. McKay, On transitive permutation groups, LMS J. Comput. Math., 1 (1998), 1-8. · Zbl 0920.20001
[7] F. Diaz y Diaz and M. Olivier, Imprimitive ninth-degree number fields with small discriminants, Math. Comp. 64 (1995), 305-321. · Zbl 0819.11070
[8] Y. Eichenlaub, Problèmes effectifs de théorie de Galois en degrés \(8\) à \(11\), Thèse, Université Bordeaux I, 1996.
[9] Y. Eichenlaub and M. Olivier, Computation of Galois groups for polynomials with degree up to eleven (1996), Preprint.
[10] F. Gassmann, Bemerkungen zur Vorstehenden Arbeit von Hurwitz, Math. Z. 25 (1926), 665-675.
[11] Helmut Hasse, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil I: Klassenkörpertheorie. Teil Ia: Beweise zu Teil I. Teil II: Reziprozitätsgesetz, Dritte Auflage, Physica-Verlag, Würzburg-Vienna, 1970 (German).
[12] Kevin Hutchinson, Norm groups of global fields, J. Number Theory 50 (1995), no. 2, 323 – 328. · Zbl 0822.11074 · doi:10.1006/jnth.1995.1026
[13] S.-H. Kwon, Sur les discriminants minimaux des corps quaternioniens, Arch. Math. (Basel) 67 (1996), no. 2, 119 – 125 (French). · Zbl 0854.11057 · doi:10.1007/BF01268925
[14] Robert Perlis, On the equation \?_{\?}(\?)=\?_{\?’}(\?), J. Number Theory 9 (1977), no. 3, 342 – 360. · Zbl 0389.12006 · doi:10.1016/0022-314X(77)90070-1
[15] M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory, Encyclopedia of Mathematics and its Applications, vol. 30, Cambridge University Press, Cambridge, 1989. · Zbl 0685.12001
[16] Xavier-François Roblot, Unités de Stark et corps de classes de Hilbert, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 11, 1165 – 1168 (French, with English and French summaries). · Zbl 0871.11080
[17] Bart de Smit and Robert Perlis, Zeta functions do not determine class numbers, Bull. Amer. Math. Soc. (N.S.) 31 (1994), no. 2, 213 – 215. · Zbl 0814.11053
[18] G. W. Smith, Some polynomials over \(\mathbb Q(t)\) and their Galois groups (1993), Preprint.
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