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1999 | 89 | 4 | 379-396
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Index form equations in quintic fields

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The problem of determining power integral bases in algebraic number fields is equivalent to solving the corresponding index form equations. As is known (cf. Győry [25]), every index form equation can be reduced to an equation system consisting of unit equations in two variables over the normal closure of the original field. However, the unit rank of the normal closure is usually too large for practical use. In a recent paper Győry [27] succeeded in reducing index form equations to systems of unit equations in which the unknown units are elements of unit groups generated by much fewer generators. On the other hand, Wildanger [32] worked out an efficient enumeration algorithm that makes it feasible to solve unit equations even if the rank of the unit group is ten. Combining these developments we describe an algorithm to solve completely index form equations in quintic fields. The method is illustrated by numerical examples: we computed all power integral bases in totally real quintic fields with Galois group S₅.
Czasopismo
Rocznik
Tom
89
Numer
4
Strony
379-396
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-11-27
poprawiono
1999-01-14
Twórcy
  • Mathematical Institute, Kossuth Lajos University, H-4010 Debrecen, Pf.12, Hungary
  • Mathematical Institute, Kossuth Lajos University, H-4010 Debrecen, Pf.12, Hungary
Bibliografia
  • [1] A. Baker and G. Wüstholz, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19-62.
  • [2] H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993.
  • [3] M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner and K. Wildanger, KANT V4, J. Symbolic Comput. 24 (1997), 267-283.
  • [4] J. H. Evertse and K. Győry, Decomposable form equations, in: New Advances in Transcendence Theory, A. Baker (ed.), Cambridge Univ. Press, 1988, 175-202.
  • [5] U. Fincke and M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comp. 44 (1985), 463-471.
  • [6] I. Gaál, Inhomogeneous discriminant form equations and integral elements with given discriminant over finitely generated integral domains, Publ. Math. Debrecen 34 (1987), 109-122.
  • [7] I. Gaál, Power integral bases in orders of families of quartic fields, ibid. 42 (1993), 253-263.
  • [8] I. Gaál, Computing all power integral bases in orders of totally real cyclic sextic number fields, Math. Comp. 65 (1996), 801-822.
  • [9] I. Gaál, Computing elements of given index in totally complex cyclic sextic fields, J. Symbolic Comput. 20 (1995), 61-69.
  • [10] I. Gaál, Power integral bases in algebraic number fields, Proc. Conf. Mátraháza, 1995, Ann. Univ. Sci. Budapest Eötvös Sect. Comp., to appear.
  • [11] I. Gaál, Application of Thue equations to computing power integral bases in algebraic number fields, in: Algorithmic Number Theory (Talence, 1996), H. Cohen (ed.), Lecture Notes in Comput. Sci. 1122, Springer, 1996, 151-155.
  • [12] I. Gaál, Power integral bases in composits of number fields, Canad. Math. Bull. 41 (1998), 158-165.
  • [13] I. Gaál, Power integral bases in algebraic number fields, in: Number Theory, Walter de Gruyter, 1998, 243-254.
  • [14] I. Gaál, Solving index form equations in fields of degree nine with cubic subfields, to appear.
  • [15] I. Gaál, A. Pethő and M. Pohst, On the resolution of index form equations in biquadratic number fields, I, J. Number Theory 38 (1991), 18-34.
  • [16] I. Gaál, A. Pethő and M. Pohst, On the resolution of index form equations in biquadratic number fields, II, ibid. 38 (1991), 35-51.
  • [17] I. Gaál, A. Pethő and M. Pohst, On the resolution of index form equations in biquadratic number fields, III. The bicyclic biquadratic case, ibid. 53 (1995), 100-114.
  • [18] I. Gaál, A. Pethő and M. Pohst, On the resolution of index form equations in quartic number fields, J. Symbolic Comput. 16 (1993), 563-584.
  • [19] I. Gaál, A. Pethő and M. Pohst, Simultaneous representation of integers by a pair of ternary quadratic forms - with an application to index form equations in quartic number fields, J. Number Theory 57 (1996), 90-104.
  • [20] I. Gaál and M. Pohst, On the resolution of index form equations in sextic fields with an imaginary quadratic subfield, J. Symbolic Comput. 22 (1996), 425-434.
  • [21] I. Gaál and M. Pohst, Power integral bases in a parametric family of totally real quintics, Math. Comp. 66 (1997), 1689-1696.
  • [22] I. Gaál and M. Pohst, On the resolution of relative Thue equations, to appear.
  • [23] I. Gaál and N. Schulte, Computing all power integral bases of cubic number fields, Math. Comp. 53 (1989), 689-696.
  • [24] M. N. Gras, Non monogénéité de l'anneau des entiers des extensions cycliques de ℚ de degré premier l ≥ 5, J. Number Theory 23 (1986), 347-353.
  • [25] K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné, III, Publ. Math. Debrecen 23 (1976), 141-165.
  • [26] K. Győry, On norm form, discriminant form and index form equations, in: Topics in Classical Number Theory, Colloq. Math. Soc. János Bolyai 34, North-Holland, 1984, 617-676.
  • [27] K. Győry, Bounds for the solutions of decomposable form equations, Publ. Math. Debrecen 52 (1998), 1-31.
  • [28] K. Győry, Recent bounds for the solutions of decomposable form equations, in: Number Theory, Walter de Gruyter, 1998, 255-270.
  • [29] M. Klebel, Zur Theorie der Potenzganzheitsbases bei relativ galoisschen Zahlkörpern, Dissertation, Univ. Augsburg, 1995.
  • [30] D. Koppenhöfer, Über projektive Darstellungen von Algebren kleinen Ranges, Dissertation, Univ. Tübingen, 1994.
  • [31] N. P. Smart, Solving discriminant form equations via unit equations, J. Symbolic Comput. 21 (1996), 367-374.
  • [32] K. Wildanger, Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern mit einer Anwendung auf die Bestimmung aller ganzen Punkte einer Mordellschen Kurve, Dissertation, Technical University, Berlin, 1997.
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Bibliografia
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