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Computing all elements of given index in sextic fields with a cubic subfield. (English) Zbl 1081.11022

The author develops a method for the computation of all elements of given index in sextic fields containing a cubic subfield. He follows well known ideas [see I. Gaál, Diophantine equations and power integral bases (Birkhäuser, Boston) (2002; Zbl 1016.11059) for a survey]. As usual, he transforms the index form equation into a unit equation. In the worst case, this requires to solve such an equation over a totally real field of Galois group \(S_4 \times C_2\) which contains 11 fundamental units. One example for an index 1 is included.

MSC:

11D57 Multiplicative and norm form equations
11Y50 Computer solution of Diophantine equations
11Y40 Algebraic number theory computations
11R21 Other number fields

Citations:

Zbl 1016.11059

Software:

KANT/KASH
PDFBibTeX XMLCite
Full Text: EuDML

References:

[1] A. Baker, G. Wüstholz: Logarithmic forms and group varieties. J. Reine Angew. Math., 442 (1993), 19-62. · Zbl 0788.11026
[2] M. Daberkow C. Fieker J. Klüners M. Pohst K. Roegner, K. Wildanger: KANT V4. J. Symbolic Comput., 24 (1997), 267-283. · Zbl 0886.11070 · doi:10.1006/jsco.1996.0126
[3] U. Fincke, M. Pohst: Improved methods for calculating vectors of short lenght in a lattice, including a complexity analysis. Math. Comput., 44 (1985), 463-471. · Zbl 0556.10022 · doi:10.2307/2007966
[4] I. Gaál: Computing elements of given index in totally complex cyclic sextic fields. J.Symbolic Comput., 20 (1995), 61-69. · Zbl 0857.11068 · doi:10.1006/jsco.1995.1038
[5] I. Gaál: Computing all power integral bases in orders of totally real cycłic sextic number fìelds. Math. Comput., 65 (1996), 801-822. · Zbl 0857.11069 · doi:10.1090/S0025-5718-96-00708-9
[6] I. Gaál: Diophantine equations and power integral bases. Birkhäuser, Boston, 2002. · Zbl 1016.11059
[7] I. Gaál, K. Györy: On the resolution of index form equations in quintic fìelds. Acta Arith., 89 (1999), 379-396.
[8] I. Gaál, M. Pohst: On the resolution of index form equations in sextic fìelds with an imaginary quadratic subfìeld. J.Symbolic Comput., 22 (1996), 425-434. · Zbl 0873.11025 · doi:10.1006/jsco.1996.0060
[9] I. Gaál, M. Pohst: On the resolution of relative Thue equations. Math. Comput., 71 (2002), 429-440. · Zbl 0985.11070 · doi:10.1090/S0025-5718-01-01329-1
[10] K. Györy: Sur les polynomes a coefficients entiers et de discriminant donne, III. Publ. Math. (Debrecen), 23 (1976), 141-165. · Zbl 0354.10041
[11] K. Győry: Discriminant form and index form equations. Algebraic number theory and diophantine analysis, Proc. Conf. Graz 1998, Walter de Gruyter 2000, 191-214. · Zbl 0962.11020
[12] I. Járasi: Power integral bases in sextic fìelds with a cubic subfìeld. Acta Sci. Math. Szeged, to appear.
[13] K. Wildanger: Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern mit einer Anwendung auf die Bestimmung aller ganzen Punkte erner Mordellschen Kurve. Dissertation, Technical University, Berlin) · Zbl 0912.11061
[14] K. Wildanger: Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern. J.Number Theory, 82 (2000), 188-224. · Zbl 0952.11032 · doi:10.1006/jnth.1999.2414
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