Calculating Power Integral bases by Solving Relative Thue Equations

Gaál, István; Remete, László; Szabó, Tímea

doi:10.2478/tmmp-2014-0020

Abstract

In our recent paper I. Gaál: Calculating “small” solutions of relative Thue equations, J. Experiment. Math. (to appear) we gave an efficient algorithm to calculate “small” solutions of relative Thue equations (where “small” means an upper bound of type 10⁵⁰⁰ for the sizes of solutions). Here we apply this algorithm to calculating power integral bases in sextic fields with an imaginary quadratic subfield and to calculating relative power integral bases in pure quartic extensions of imaginary quadratic fields. In both cases the crucial point of the calculation is the resolution of a relative Thue equation. We produce numerical data that were not known before.

References

[1] BAKER, A.: Transcendental number theory. Cambridge University Press, Cambridge, 1990.
Search in Google Scholar
[2] CHAR, B.W.-GEDDES, K. O.-GONNET, G.H.-MONAGAN, M. B.-WATT, S. M.: MAPLE, Reference Manual. Watcom Publications, Waterloo, Canada, 1988.
Search in Google Scholar
[3] GAÁL, I.: Application of Thue equations to computing power integral bases in algebraic number fields, in: Proc. of the 2nd Internat. Symposium on Algorithmic Number Theory-ANTS-II (H. Cohen, ed.), Talence, France, 1996, Lecture Notes in Computer Sci., Vol. 1122, Springer-Verlag, Berlin 1996, pp. 151-155.
Search in Google Scholar
[4] GAÁL, I.: Power integral bases in composits of number fields, Canad. Math. Bull. 41 (1998), 158-161.10.4153/CMB-1998-025-3
Search in Google Scholar
[5] GAÁL, I.: Solving index form equations in fields of degree nine with cubic subfields, J. Symbolic Comput. 30 (2000), 181-193.10.1006/jsco.1999.0356
Search in Google Scholar
[6] GAÁL, I.: Power integral bases in cubic relative extensions, Experiment. Math. 10 (2001), 133-139.10.1080/10586458.2001.10504436
Search in Google Scholar
[7] GAÁL, I.: Diophantine equations and power integral bases, Boston, Birkhäuser, 2002.10.1007/978-1-4612-0085-7
Search in Google Scholar
[8] GAÁL, I.: Calculating “small” solutions of relative Thue equations, J. Experiment. Math. (to appear).
Search in Google Scholar
[9] GAÁL, I.-POHST, M.: On the resolution of index form equations in sextic fields with an imaginary quadratic subfield, J. Symbolic Comput. 22 (1996), 425-434.10.1006/jsco.1996.0060
Search in Google Scholar
[10] GAÁL, I.-POHST, M.: On the resolution of index form equations in relative quartic extensions, J. Number Theory 85 (2000), 201-219.10.1006/jnth.2000.2541
Search in Google Scholar
[11] GAÁL, I.-POHST, M.: On the resolution of relative Thue equations, Math. Comput. 71 (2002), 429-440.10.1090/S0025-5718-01-01329-1
Search in Google Scholar
[12] GAÁL, I.-PETHŐ, A.-POHST, M.: 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), 90-104.10.1006/jnth.1996.0035
Search in Google Scholar
[13] GAÁL, I.-REMETE, L.: Binomial Thue equations and power integral bases in pure quartic fields, JP J. Algebra Number Theory Appl. 32 (2014), 49-61.
Search in Google Scholar
[14] GAÁL, I.-SZABÓ, T.: Relative power integral bases in infinite families of quartic extensions of quadratic field, JP J. Algebra Number Theory Appl. 29 (2013), 31-43.
Search in Google Scholar
[15] HAMEED, A.-NAKAHARA, T.-HUSNINE, S. M.-AHMAD, S.: On the existence of canonical number system in certain classes of pure algebraic number fields, J. Prime Res. Math. 7 (2011), 19-24.
Search in Google Scholar
[16] LENSTRA, A. K.-LENSTRA, H. W., JR.-LOVÁSZ, L.: Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515-534.10.1007/BF01457454
Search in Google Scholar
[17] NARKIEWICZ, W.: Elementary and Analytic Theory of Algebraic Numbers (2nd ed.). Springer-Verlag, Berlin, 1974.
Search in Google Scholar
[18] OLIVIER, M.: Corps sextiques contenant un corps quadratique (I), Sémin. Théor. Nombres Bordx., Sér. II 1 (1989), 205-250.
Search in Google Scholar
[19] PETHŐ, A.: On the resolution of Thue inequalities, J. Symbolic Comput. 4 (1987), 103-109.10.1016/S0747-7171(87)80059-7
Search in Google Scholar

Calculating Power Integral bases by Solving Relative Thue Equations

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