Computing all power integral bases in orders of totally real cyclic sextic number fields

Gaál, István

doi:10.1090/S0025-5718-96-00708-9

Computing all power integral bases in orders of totally real cyclic sextic number fields
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Math. Comp. 65 (1996), 801-822 Request permission

Abstract:

An algorithm is given for determining all power integral bases in orders of totally real cyclic sextic number fields. The orders considered are in most cases the maximal orders of the fields. The corresponding index form equation is reduced to a relative Thue equation of degree 3 over the quadratic subfield and to some inhomogeneous Thue equations of degree 3 over the rationals. At the end of the paper, numerical examples are given.

References

A. Baker and H. Davenport, The equations $3x^{2}-2=y^{2}$ and $8x^{2}-7=z^{2}$, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129–137. MR 248079, DOI 10.1093/qmath/20.1.129
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
Josef Blass, A. M. W. Glass, David K. Manski, David B. Meronk, and Ray P. Steiner, Constants for lower bounds for linear forms in the logarithms of algebraic numbers. II. The homogeneous rational case, Acta Arith. 55 (1990), no. 1, 15–22. MR 1056111, DOI 10.4064/aa-55-1-15-22
Josef Blass, A. M. W. Glass, David K. Manski, David B. Meronk, and Ray P. Steiner, Corrigendum to the paper: “Constants for lower bounds for linear forms in the logarithms of algebraic numbers. II. The homogeneous rational case” [Acta Arith. 55 (1990), no. 1, 15–22; MR1056111 (91h:11064)], Acta Arith. 65 (1993), no. 4, 383. MR 1259345
W. J. Ellison, Recipes for solving Diophantine problems by Baker’s method, Séminaire de Théorie des Nombres, 1970-1971 (Univ. Bordeaux I, Talence), Exp. No. 11, Lab. Théorie des Nombres, Centre Nat. Recherche Sci., Talence, 1971, pp. 10. MR 0389760
V. Ennola, S. Mäki, and R. Turunen, On real cyclic sextic fields, Math. Comp. 45 (1985), no. 172, 591–611. MR 804948, DOI 10.1090/S0025-5718-1985-0804948-5
J.-H. Evertse, K. Győry, C. L. Stewart, and R. Tijdeman, $S$-unit equations and their applications, New advances in transcendence theory (Durham, 1986) Cambridge Univ. Press, Cambridge, 1988, pp. 110–174. MR 971998
U. Fincke and M. Pohst, A procedure for determining algebraic integers of given norm, Computer algebra (London, 1983) Lecture Notes in Comput. Sci., vol. 162, Springer, Berlin, 1983, pp. 194–202. MR 774811, DOI 10.1007/3-540-12868-9_{1}03
István Gaál, On the resolution of inhomogeneous norm form equations in two dominating variables, Math. Comp. 51 (1988), no. 183, 359–373. MR 942162, DOI 10.1090/S0025-5718-1988-0942162-6
I. Gaál, A. Pethö, and M. Pohst, On the resolution of index form equations in biquadratic number fields. I, II, J. Number Theory 38 (1991), no. 1, 18–34, 35–51. MR 1105669, DOI 10.1016/0022-314X(91)90090-X
I. Gaál, A. Pethö, and M. Pohst, On the resolution of index form equations in biquadratic number fields. I, II, J. Number Theory 38 (1991), no. 1, 18–34, 35–51. MR 1105669, DOI 10.1016/0022-314X(91)90090-X
I. Gaál, A. Pethő and M. Pohst, On the resolution of index form equations in biquadratic number fields, III. The bicyclic biquadratic case, J. Number Theory 53 (1995), 100–114.
I. Gaál, A. Pethő and M. Pohst, On the resolution of index form equations in dihedral number fields, J. Experimental Math. 3 (1994), 245–254.
I. Gaál, A. Pethő and M. Pohst, On the resolution of index form equations, Proc. of the 1991 International Symposium on Symbolic and Algebraic Computation, ed. by Stephen M. Watt, ACM Press, 1991, pp. (185-186).
István Gaál, Attila Pethő, and Michael Pohst, On the resolution of index form equations in quartic number fields, J. Symbolic Comput. 16 (1993), no. 6, 563–584. MR 1279534, DOI 10.1006/jsco.1993.1064
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, to appear, J. Number Theory.
I. Gaál and N. Schulte, Computing all power integral bases of cubic fields, Math. Comp. 53 (1989), no. 188, 689–696. MR 979943, DOI 10.1090/S0025-5718-1989-0979943-X
K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné, Acta Arith. 23 (1973), 419–426. MR 437489, DOI 10.4064/aa-23-4-419-426
A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515–534. MR 682664, DOI 10.1007/BF01457454
Sirpa Mäki, The determination of units in real cyclic sextic fields, Lecture Notes in Mathematics, vol. 797, Springer, Berlin, 1980. MR 584794, DOI 10.1007/BFb0088938
Ivan Niven and Herbert S. Zuckerman, An introduction to the theory of numbers, 4th ed., John Wiley & Sons, New York-Chichester-Brisbane, 1980. MR 572268
Michael E. Pohst, Computational algebraic number theory, DMV Seminar, vol. 21, Birkhäuser Verlag, Basel, 1993. MR 1243639, DOI 10.1007/978-3-0348-8589-8
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
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
Wolfgang M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR 568710
V. G. Sprindžuk, Representation of numbers by the norm forms with two dominating variables, J. Number Theory 6 (1974), 481–486. MR 354567, DOI 10.1016/0022-314X(74)90044-4
N. Tzanakis and B. M. M. de Weger, On the practical solution of the Thue equation, J. Number Theory 31 (1989), no. 2, 99–132. MR 987566, DOI 10.1016/0022-314X(89)90014-0
N. Tzanakis and B. M. M. de Weger, How to explicitly solve a Thue-Mahler equation, Compositio Math. 84 (1992), no. 3, 223–288. MR 1189890
B. M. M. de Weger, Algorithms for Diophantine equations, CWI Tract, vol. 65, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989. MR 1026936
B. M. M. de Weger, A Thue equation with quadratic integers as variables, Math. Comp. 64 (1995), no. 210, 855–861. MR 1277766, DOI 10.1090/S0025-5718-1995-1277766-0