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A generalization of simplest number fields and their integral basis


An integral basis of the simplest number fields of degrees 3, 4 and 6 over \(\mathbb{Q}\) is well-known, and widely investigated. We generalize the simplest number fields to any degree, and show that an integral basis of these fields is repeating periodically.

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  1. Cohn, H.: A device for generating fields of even class number. Proc. Amer. Math. Soc. 7, 595–598 (1956)

    Article  MathSciNet  Google Scholar 

  2. V. Ennola, Cubic number fields with exceptional units, in: Computational Number Theory, Proc. Colloq. Debrecen, 1989 (1991), 103–128

  3. Ennola, V.: Fundamental units in a family of cubic fields. J. Théor. Nombres Bordeaux 16, 569–575 (2004)

    Article  MathSciNet  Google Scholar 

  4. Foster, K.: HT90 and "simplest" number fields. Illinois J. Math. 55, 1621–1655 (2011)

    Article  MathSciNet  Google Scholar 

  5. I. Gaál, Diophantine Equations and Power Integral Bases. Theory and Algorithms, 2nd ed., Birkhäuser (2019)

  6. Gaál, I., Jadrijević, B., Remete, L.: Simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. Int. J. Number Theory 15, 11–27 (2019)

    Article  MathSciNet  Google Scholar 

  7. Gaál, I., Remete, L.: Integral bases and monogenity of pure fields. J. Number Theory 173, 129–146 (2017)

    Article  MathSciNet  Google Scholar 

  8. Gaál, I., Remete, L.: Integral bases and monogenity of the simplest sextic fields. Acta Arith. 182, 173–183 (2018)

    Article  MathSciNet  Google Scholar 

  9. M. N. Gras, Table numérique du nombre de classes et des unités des extensions cycliques réelles de degré 4 de \({\mathbb{Q}}\), Publ. Math. Fac. Sci. Besançon, Théor. Nombres, 2 (1977-1978), 1–79

  10. M. N. Gras, Families of units in real cyclic extensions of \({\mathbb{Q}}\) of degree 6, Publ. Math. Fac. Sci. Besançon, Théor. Nombres, 1984/85-1985/86 (1986), Exp. No. 2, 27 pp

  11. Heuberger, C.: All solutions to Thomas' family of Thue equations over imaginary quadratic number fields. J. Symbolic Comput. 41, 980–998 (2006)

    Article  MathSciNet  Google Scholar 

  12. Hoshi, A.: On correspondence between solutions of a family of cubic Thue equations andisomorphism classes of the simplest cubic fields. J. Number Theory 131, 2135–2150 (2011)

    Article  MathSciNet  Google Scholar 

  13. A. Hoshi, On the simplest quartic fields and related Thue equations, in: Computer Mathematics, 9th Asian symposium, ASCM 2009, Fukuoka, Japan, December 14-17, 2009, 10th Asian symposium, ASCM 2012, Beijing, China, October 26-28, 2012, Springer (2014), pp. 67–85

  14. Hoshi, A.: On the simplest sextic fields and related Thue equations. Funct. Approx. Comment. Math. 47, 35–49 (2012)

    Article  MathSciNet  Google Scholar 

  15. Hoshi, A.: Complete solutions to a family of Thue equations of degree 12. J. Théor. Nombres Bordeaux 29, 549–568 (2017)

    Article  MathSciNet  Google Scholar 

  16. H. K. Kim and J. H. Lee, Evaluation of the Dedekind zeta function at s =-1 of the simplest quartic fields, Trends in Math., New Ser., Inf. Center for Math. Sci., 11 (2009), 63–79

  17. Konvalina, J., Liu, Y.-H.: Arithmetic progression sums of binomial coefficients. Appl. Math. Lett. 10, 11–13 (1997)

    Article  MathSciNet  Google Scholar 

  18. Lazarus, A.J.: On the class number and unit index of simplest quartic fields. Nagoya Math. J. 121, 1–13 (1991)

    Article  MathSciNet  Google Scholar 

  19. Lettl, G., Pethő, A.: Complete solution of a family of quartic Thue equations. Abh. Math. Semin. Univ. Hamb. 65, 365–383 (1995)

    Article  MathSciNet  Google Scholar 

  20. G. Lettl, A. Pethő and P. Voutier, On the arithmetic of simplest sextic fields and related Thue equations, in: Number Theory: Diophantine, Computational and Algebraic Aspects, (K. Győry, A. Pethő and V.T. Sós, eds.), Walter de Gruyter Publ. Co. (1998), 331–348

  21. Lettl, G., Pethő, A., Voutier, P.: Simple families of Thue inequalities. Trans. Amer. Math. Soc. 351, 1871–1894 (1999)

    Article  MathSciNet  Google Scholar 

  22. M. Mignotte, Verification of a conjecture of E. Thomas, J. Number Theory, 44 (1993), 172–177

  23. Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers, 3rd edn. Springer, Springer Monogr. Math. (2004)

    Book  Google Scholar 

  24. Pohst, M., Zassenhaus, H.: Algorithmic Algebraic Number Theory. Cambridge University Press (1989)

  25. Ramus, C.: Solution générale d'un problème d'analyse combinatoire. J. Reine Angew. Math. 11, 353–355 (1834)

    MathSciNet  Google Scholar 

  26. Remete, L.: Integral basis of pure fields with square-free parameter. Studia Sci. Math. Hungar. 57, 91–115 (2020)

    MathSciNet  MATH  Google Scholar 

  27. Shanks, D.: The simplest cubic fields. Math. Comput. 28, 1137–1152 (1974)

    Article  MathSciNet  Google Scholar 

  28. Thomas, E.: Complete solutions to a family of cubic diophantine equations. J. Number Theory 34, 235–250 (1990)

    Article  MathSciNet  Google Scholar 

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Correspondence to L. Remete.

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Supported through the ÚNKP-19-3 New National Excellence Program of the Ministry for Innovation and Technology.

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Remete, L. A generalization of simplest number fields and their integral basis. Acta Math. Hungar. 163, 437–461 (2021).

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