Tokyo Journal of Mathematics

Toward Noether's Problem for the Fields of Cross-ratios


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In this article, we consider an analogue of Noether's problem for the fields of cross-ratios, and discuss on a rationality problem which connects this with Noether's problem. We show that the affirmative answer of the analogue implies the affirmative answer for Noether's Problem for any permutation group with odd degree. We also obtain some negative results for various permutation groups with even degree.

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Tokyo J. Math., Volume 39, Number 3 (2017), 901-922.

First available in Project Euclid: 6 April 2017

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Zentralblatt MATH identifier

Primary: 11E04: Quadratic forms over general fields
Secondary: 11R32: Galois theory 11R34: Galois cohomology [See also 12Gxx, 19A31] 12F12: Inverse Galois theory 12F20: Transcendental extensions 14E08: Rationality questions [See also 14M20] 20B30: Symmetric groups 20B35: Subgroups of symmetric groups


TSUNOGAI, Hiroshi. Toward Noether's Problem for the Fields of Cross-ratios. Tokyo J. Math. 39 (2017), no. 3, 901--922. doi:10.3836/tjm/1491465735.

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  • H. Ahmad, M. Hajja and M.-C. Kang, Rationality of some projective linear actions, J. Algebra 228 (2000), 643–658.
  • G. Butler, The transitive groups of degree fourteen and fifteen, J. Symbolic Comput. 16 (1993), no. 5, 413–422.
  • G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), no. 8, 863–911.
  • J. K. Deveney and J. Yanik, Nonrational fixed fields, Pacific J. Math. 139 (1989), no. 1, 45–51.
  • The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.4; 2016. (
  • M. Hajja, M.-C. Kang and J. Ohm, Function fields of conics as invariant subfields, J. Algebra 163 (1994), 383–403.
  • K. Hashimoto and H. Tsunogai, Generic polynomials over $\Q$ with two parameters for the transitive groups of degree five, Proc. Japan Acad. 79A (2003), 148–151.
  • K. Hashimoto and H. Tsunogai, Noether's problem for transitive permutation groups of degree 6, Adv. Stud. Pure Math. 63 (Galois-Teichmüller Theory and Arithmetic Geometry) (2012), 189–220.
  • A. Hoshi, M.-C. Kang and H. Kitayama, Quasi-monomial actions and some $4$-dimensional rationality problems, J. Algebra 403 (2014), 363–400.
  • G. Kemper and E. Mattig, Generic polynomials with few parameters, J. Symbolic Computation 30 (2000), 843–857.
  • T. Maeda, Noether's Problem for $A_5$, J. Algebra 125 (1989), 418–430.
  • T. Miyata, Invariants of certain groups I, Nagoya Math. J. 41 (1971), 69–73.
  • E. Noether, Rationale Funktionenkörper, Jahresbericht der Dt.Math.-Verein. 22 (1913), 316–319.
  • E. Noether, Gleichungen mit vorgeschriebener Gruppe, Math. Ann. 78 (1916), 221–229.
  • G. F. Royle, The transitive groups of degree twelve, J. Symbolic Comput. 4 (1987), no. 2, 255–268.
  • J. P. Serre, Local fields, Graduate Texts in Mathematics 67, Springer-Verlag, 1979.
  • J. P. Serre, Cohomologie galoisienne (Fifth edition), Lecture Notes in Mathematics 5, Springer-Verlag, 1994.
  • D. D. Triantaphyllou, Invariants of finite groups acting non-linearly on rational function fields, J. Pure Appl. Algebra 18 (1980), 315–331.
  • M. Watanabe, Relative rationality of field extensions related to Noether's Problem with respect to the alternating group of degree $6$ (in Japanese), Master's thesis, Sophia University, 2008.