Functiones et Approximatio Commentarii Mathematici

On the simplest sextic fields and related Thue equations

Akinari Hoshi

Full-text: Open access


We consider the parametric family of sextic Thue equations $$x^6-2mx^5y-5(m+3)x^4y^2-20x^3y^3+5mx^2y^4+2(m+3)xy^5+y^6=\lambda$$ where $m\in\mathbb{Z}$ is an integer and $\lambda$ is a divisor of $27(m^2+3m+9)$. We show that the only solutions to the equations are the trivial ones with $xy(x+y)(x-y)(x+2y)(2x+y)=0$.

Article information

Funct. Approx. Comment. Math., Volume 47, Number 1 (2012), 35-49.

First available in Project Euclid: 25 September 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D41: Higher degree equations; Fermat's equation
Secondary: 11D59: Thue-Mahler equations 11R20: Other abelian and metabelian extensions 11Y40: Algebraic number theory computations 12F10: Separable extensions, Galois theory

sextic Thue equations simplest sextic fields field isomorphism problem multi-resolvent polynomial


Hoshi, Akinari. On the simplest sextic fields and related Thue equations. Funct. Approx. Comment. Math. 47 (2012), no. 1, 35--49. doi:10.7169/facm/2012.47.1.3.

Export citation


  • C. Adelmann, The decomposition of primes in torsion point fields, Lecture Notes in Mathematics, 1761, Springer-Verlag, Berlin, 2001.
  • R. J. Chapman, Automorphism polynomials in cyclic cubic extensions, J. Number Theory 61 (1996), 283–291.
  • H. Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, 138, Springer-Verlag, Berlin, 1993.
  • H. Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, 193, Springer-Verlag, New York, 2000.
  • V. Ennola, Cubic number fields with exceptional units, Computational number theory (Debrecen, 1989), 103–128, de Gruyter, Berlin, 1991.
  • I. Gaál, Diophantine equations and power integral bases. New computational methods, Birkhäuser Boston, Inc., Boston, MA, 2002.
  • The GAP Group, GAP –- Groups, Algorithms, and Programming, Version 4.4.10; 2007 (
  • K. Girstmair, On the computation of resolvents and Galois groups, Manuscripta Math. 43 (1983), 289–307.
  • M.N. Gras, Familles d'unités dans les extensions cycliques réelles de degré $6$ de $Q$, (French) Théorie des nombres, Années 1984/85–1985/86, Fasc. 2, Exp. No. 2, 27 pp., Publ. Math. Fac. Sci. Besançon, Univ. Franche-Comté, Besançon, 1986.
  • M.N. Gras, Special units in real cyclic sextic fields, Math. Comp. 48 (1987), 179–182.
  • M. Hajja, M. Kang, Some actions of symmetric groups, J. Algebra 177 (1995), 511–535.
  • K. Hashimoto, A. Hoshi, Families of cyclic polynomials obtained from geometric generalization of Gaussian period relations, Math. Comp. 74 (2005), 1519–1530.
  • K. Hashimoto, K. Miyake, Inverse Galois problem for dihedral groups, Number theory and its applications (Kyoto, 1997), 165–181, Dev. Math., 2, Kluwer Acad. Publ., Dordrecht, 1999.
  • A. Hoshi, On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields, J. Number Theory 131 (2011), 2135–2150.
  • A. Hoshi, On the simplest quartic fields and related Thue equations, preprint, arXiv:1004.1960v2.
  • A. Hoshi, K. Miyake, Tschirnhausen transformation of a cubic generic polynomial and a $2$-dimensional involutive Cremona transformation, Proc. Japan Acad. Ser. A 83 (2007), 21–26.
  • A. Hoshi, K. Miyake, A geometric framework for the subfield problem of generic polynomials via Tschirnhausen transformation, Number theory and applications, 65–104, Hindustan Book Agency, New Delhi, 2009.
  • A. Hoshi, K. Miyake, On the field intersection problem of quartic generic polynomials via formal Tschirnhausen transformation, Comment. Math. Univ. St. Pauli 58 (2009), 51–86.
  • A. Hoshi, K. Miyake, On the field intersection problem of generic polynomials: a survey, RIMS Kôkyûroku Bessatsu B12 (2009), 231–247.
  • A. Hoshi, K. Miyake, Some Diophantine problems arising from the isomorphism problem of generic polynomials, Number Theory: Dreaming in Dreams, 87–105, Proceedings of the 5th China-Japan Seminar, World Sci. Publ., Singapore, 2010.
  • A. Hoshi, K. Miyake, A note on the field isomorphism problem of $X^3+sX+s$ and related cubic Thue equations, Interdiscip. Inform. Sci. 16 (2010), 45–54.
  • A. Hoshi, K. Miyake, On the field intersection problem of solvable quintic generic polynomials, Int. J. Number Theory 6 (2010), 1047–1081.
  • M. Kida, Kummer theory for norm algebraic tori, J. Algebra 293 (2005), 427–447.
  • T. Komatsu, Arithmetic of Rikuna's generic cyclic polynomial and generalization of Kummer theory, Manuscripta Math. 114 (2004), 265–279.
  • S. Lang, Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften, 231, Springer-Verlag, Berlin-New York, 1978.
  • S. Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983.
  • G. Lettl, A. Pethö, P. Voutier, On the arithmetic of simplest sextic fields and related Thue equations, Number theory (Eger, 1996), 331–348, de Gruyter, Berlin, 1998.
  • G. Lettl, A. Pethö, P. Voutier, Simple families of Thue inequalities, Trans. Amer. Math. Soc. 351 (1999), 1871–1894.
  • S. R. Louboutin, Efficient computation of root numbers and class numbers of parametrized families of real abelian number fields, Math. Comp. 76 (2007), 455–473.
  • K. Miyake, Linear fractional transformations and cyclic polynomials, Algebraic number theory (Hapcheon/Saga, 1996), Adv. Stud. Contemp. Math. (Pusan) 1 (1999), 137–142.
  • P. Morton, Characterizing cyclic cubic extensions by automorphism polynomials, J. Number Theory 49 (1994), 183–208.
  • H. Ogawa, Quadratic reduction of multiplicative group and its applications, (Japanese) Algebraic number theory and related topics (Kyoto, 2002), Sūrikaisekikenkyūsho Kōkyūroku No. 1324, 217–224, 2003.
  • R. Okazaki, Geometry of a cubic Thue equation, Publ. Math. Debrecen 61 (2002), 267–314.
  • R. Okazaki, The simplest cubic fields are non-isomorphic to each other, presentation sheet, available from$^\text{\texttildelow}$rokazaki/papers.html.
  • D. Poulakis, E. Voskos, On the practical solution of genus zero Diophantine equations, J. Symbolic Comput. 30 (2000), 573–582.
  • N. Rennert and A. Valibouze, Calcul de résolvantes avec les modules de Cauchy, Experiment. Math. 8 (1999), 351–366.
  • N. Rennert, A parallel multi-modular algorithm for computing Lagrange resolvens, J. Symbolic Comput. 37 (2004), 547–556.
  • Y. Rikuna, On simple families of cyclic polynomials, Proc. Amer. Math. Soc. 130 (2002), 2215–2218.
  • J. R. Sendra, F. Winkler, S. Pérez-Díaz, Rational algebraic curves. A computer algebra approach, Algorithms and Computation in Mathematics, 22. Springer, Berlin, 2008.
  • D. Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137–1152.
  • A. Togbé, On the solutions of a family of sextic Thue equations, Number theory for the millennium, III (Urbana, IL, 2000), 285–299, A K Peters, Natick, MA, 2002.
  • I. Wakabayashi, Number of solutions for cubic Thue equations with automorphisms, Ramanujan J. 14 (2007), 131–154.
  • I. Wakabayashi, Simple families of Thue inequalities, Ann. Sci. Math. Québec 31 (2007), 211–232.