Noether’s problem and $\mathbb {Q}$-generic polynomials for the normalizer of the $8$-cycle in $S_8$ and its subgroups

Authors:
Ki-ichiro Hashimoto, Akinari Hoshi and Yuichi Rikuna

Journal:
Math. Comp. **77** (2008), 1153-1183

MSC (2000):
Primary 12F12, 14E08, 11R32.

DOI:
https://doi.org/10.1090/S0025-5718-07-02094-7

Published electronically:
December 3, 2007

MathSciNet review:
2373196

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study Noether’s problem for various subgroups $H$ of the normalizer of a group $\mathbf {C}_8$ generated by an $8$-cycle in $S_8$, the symmetric group of degree $8$, in three aspects according to the way they act on rational function fields, i.e., $\mathbb {Q}(X_0,\ldots ,X_7), \mathbb {Q}(x_1,\ldots ,x_4)$, and $\mathbb {Q}(x,y)$. We prove that it has affirmative answers for those $H$ containing $\mathbf {C}_8$ properly and derive a $\mathbb {Q}$-generic polynomial with four parameters for each $H$. On the other hand, it is known in connection to the negative answer to the same problem for ${\mathbf C}_8/{\mathbb {Q}}$ that there does not exist a $\mathbb {Q}$-generic polynomial for ${\mathbf C}_8$. This leads us to the question whether and how one can describe, for a given field $K$ of characteristic zero, the set of ${\mathbf C}_8$-extensions $L/K$. One of the main results of this paper gives an answer to this question.

- J. Buhler and Z. Reichstein,
*On the essential dimension of a finite group*, Compositio Math.**106**(1997), no. 2, 159–179. MR**1457337**, DOI https://doi.org/10.1023/A%3A1000144403695 - Huah Chu, Shou-Jen Hu, and Ming-chang Kang,
*Noether’s problem for dihedral 2-groups*, Comment. Math. Helv.**79**(2004), no. 1, 147–159. MR**2031703**, DOI https://doi.org/10.1007/s00014-003-0783-8 - Frank R. DeMeyer,
*Generic polynomials*, J. Algebra**84**(1983), no. 2, 441–448. MR**723401**, DOI https://doi.org/10.1016/0021-8693%2883%2990087-X - Frank DeMeyer and Thomas McKenzie,
*On generic polynomials*, J. Algebra**261**(2003), no. 2, 327–333. MR**1966633**, DOI https://doi.org/10.1016/S0021-8693%2802%2900678-6 - Shizuo Endô and Takehiko Miyata,
*Invariants of finite abelian groups*, J. Math. Soc. Japan**25**(1973), 7–26. MR**311754**, DOI https://doi.org/10.2969/jmsj/02510007 - Skip Garibaldi, Alexander Merkurjev, and Jean-Pierre Serre,
*Cohomological invariants in Galois cohomology*, University Lecture Series, vol. 28, American Mathematical Society, Providence, RI, 2003. MR**1999383** - Mowaffaq Hajja,
*A note on monomial automorphisms*, J. Algebra**85**(1983), no. 1, 243–250. MR**723077**, DOI https://doi.org/10.1016/0021-8693%2883%2990128-X - Mowaffaq Hajja,
*Rationality of finite groups of monomial automorphisms of $k(x,y)$*, J. Algebra**109**(1987), no. 1, 46–51. MR**898335**, DOI https://doi.org/10.1016/0021-8693%2887%2990162-1 - Mowaffaq Hajja and Ming Chang Kang,
*Three-dimensional purely monomial group actions*, J. Algebra**170**(1994), no. 3, 805–860. MR**1305266**, DOI https://doi.org/10.1006/jabr.1994.1366 - Ki-ichiro Hashimoto,
*On Brumer’s family of RM-curves of genus two*, Tohoku Math. J. (2)**52**(2000), no. 4, 475–488. MR**1793932**, DOI https://doi.org/10.2748/tmj/1178207751 - Ki-Ichiro Hashimoto and Akinari Hoshi,
*Families of cyclic polynomials obtained from geometric generalization of Gaussian period relations*, Math. Comp.**74**(2005), no. 251, 1519–1530. MR**2137015**, DOI https://doi.org/10.1090/S0025-5718-05-01750-3 - Ki-ichiro Hashimoto and Akinari Hoshi,
*Geometric generalization of Gaussian period relations with application to Noether’s problem for meta-cyclic groups*, Tokyo J. Math.**28**(2005), no. 1, 13–32. MR**2149620**, DOI https://doi.org/10.3836/tjm/1244208276 - Ki-ichiro Hashimoto and Hiroshi Tsunogai,
*Generic polynomials over $\mathbf Q$ with two parameters for the transitive groups of degree five*, Proc. Japan Acad. Ser. A Math. Sci.**79**(2003), no. 9, 142–145. MR**2022057** - Akinari Hoshi,
*Noether’s problem for some meta-abelian groups of small degree*, Proc. Japan Acad. Ser. A Math. Sci.**81**(2005), no. 1, 1–6. MR**2068482** - Christian U. Jensen, Arne Ledet, and Noriko Yui,
*Generic polynomials*, Mathematical Sciences Research Institute Publications, vol. 45, Cambridge University Press, Cambridge, 2002. Constructive aspects of the inverse Galois problem. MR**1969648** - Ming-chang Kang,
*Introduction to Noether’s problem for dihedral groups*, Proceedings of the International Conference on Algebra, 2004, pp. 71–78. MR**2058965** - Ming-Chang Kang,
*Noether’s problem for dihedral 2-groups. II*, Pacific J. Math.**222**(2005), no. 2, 301–316. MR**2225074**, DOI https://doi.org/10.2140/pjm.2005.222.301 - Gregor Kemper,
*A constructive approach to Noether’s problem*, Manuscripta Math.**90**(1996), no. 3, 343–363. MR**1397662**, DOI https://doi.org/10.1007/BF02568311 - Gregor Kemper,
*Generic polynomials are descent-generic*, Manuscripta Math.**105**(2001), no. 1, 139–141. MR**1885819**, DOI https://doi.org/10.1007/s002290170015 - Gregor Kemper and Elena Mattig,
*Generic polynomials with few parameters*, J. Symbolic Comput.**30**(2000), no. 6, 843–857. Algorithmic methods in Galois theory. MR**1800681**, DOI https://doi.org/10.1006/jsco.1999.0385 - Arne Ledet,
*Generic polynomials for quasi-dihedral, dihedral and modular extensions of order $16$*, Proc. Amer. Math. Soc.**128**(2000), no. 8, 2213–2222. MR**1707525**, DOI https://doi.org/10.1090/S0002-9939-99-05570-7 - Arne Ledet,
*Finite groups of essential dimension one*, J. Algebra**311**(2007), no. 1, 31–37. MR**2309876**, DOI https://doi.org/10.1016/j.jalgebra.2006.12.027 - H. W. Lenstra Jr.,
*Rational functions invariant under a finite abelian group*, Invent. Math.**25**(1974), 299–325. MR**347788**, DOI https://doi.org/10.1007/BF01389732 - Gunter Malle and B. Heinrich Matzat,
*Inverse Galois theory*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. MR**1711577** - Takehiko Miyata,
*Invariants of certain groups. I*, Nagoya Math. J.**41**(1971), 69–73. MR**272923** - Katsuya Miyake,
*Linear fractional transformations and cyclic polynomials*, Adv. Stud. Contemp. Math. (Pusan)**1**(1999), 137–142. Algebraic number theory (Hapcheon/Saga, 1996). MR**1701914** - Emmy Noether,
*Gleichungen mit vorgeschriebener Gruppe*, Math. Ann.**78**(1917), no. 1, 221–229 (German). MR**1511893**, DOI https://doi.org/10.1007/BF01457099 - Yūichi Rikuna,
*Explicit constructions of generic polynomials for some elementary groups*, Galois theory and modular forms, Dev. Math., vol. 11, Kluwer Acad. Publ., Boston, MA, 2004, pp. 173–194. MR**2059763**, DOI https://doi.org/10.1007/978-1-4613-0249-0_9 - David J. Saltman,
*Generic Galois extensions and problems in field theory*, Adv. in Math.**43**(1982), no. 3, 250–283. MR**648801**, DOI https://doi.org/10.1016/0001-8708%2882%2990036-6 - David J. Saltman,
*Retract rational fields and cyclic Galois extensions*, Israel J. Math.**47**(1984), no. 2-3, 165–215. MR**738167**, DOI https://doi.org/10.1007/BF02760515 - Leila Schneps,
*On cyclic field extensions of degree $8$*, Math. Scand.**71**(1992), no. 1, 24–30. MR**1216101**, DOI https://doi.org/10.7146/math.scand.a-12408 - Jean-Pierre Serre,
*Topics in Galois theory*, Research Notes in Mathematics, vol. 1, Jones and Bartlett Publishers, Boston, MA, 1992. Lecture notes prepared by Henri Damon [Henri Darmon]; With a foreword by Darmon and the author. MR**1162313** - Yuan Yuan Shen,
*Unit groups and class numbers of real cyclic octic fields*, Trans. Amer. Math. Soc.**326**(1991), no. 1, 179–209. MR**1031243**, DOI https://doi.org/10.1090/S0002-9947-1991-1031243-3 - Yuan Yuan Shen and Lawrence C. Washington,
*A family of real $2^n$-tic fields*, Trans. Amer. Math. Soc.**345**(1994), no. 1, 413–434. MR**1264151**, DOI https://doi.org/10.1090/S0002-9947-1994-1264151-6 - Richard G. Swan,
*Invariant rational functions and a problem of Steenrod*, Invent. Math.**7**(1969), 148–158. MR**244215**, DOI https://doi.org/10.1007/BF01389798 - Richard G. Swan,
*Noether’s problem in Galois theory*, Emmy Noether in Bryn Mawr (Bryn Mawr, Pa., 1982) Springer, New York-Berlin, 1983, pp. 21–40. MR**713790** - V. E. Voskresenskiĭ,
*Fields of invariants of abelian groups*, Uspehi Mat. Nauk**28**(1973), no. 4(172), 77–102 (Russian). MR**0392935** - V. E. Voskresenskiĭ,
*Algebraic groups and their birational invariants*, Translations of Mathematical Monographs, vol. 179, American Mathematical Society, Providence, RI, 1998. Translated from the Russian manuscript by Boris Kunyavski [Boris È. Kunyavskiĭ]. MR**1634406**

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Additional Information

**Ki-ichiro Hashimoto**

Affiliation:
Department of Applied Mathematics, School of Fundamental Science and Engineering, Waseda University, 3–4–1 Ohkubo, Shinjuku-ku, Tokyo, 169–8555, Japan

Email:
khasimot@waseda.jp

**Akinari Hoshi**

Affiliation:
Department of Mathematics, School of Education, Waseda University, 1–6–1 Nishi-Waseda, Shinjuku-ku, Tokyo, 169–8050, Japan

MR Author ID:
714371

Email:
hoshi@ruri.waseda.jp

**Yuichi Rikuna**

Affiliation:
Department of Applied Mathematics, School of Fundamental Science and Engineering, Waseda University, 3–4–1 Ohkubo, Shinjuku-ku, Tokyo, 169–8555, Japan

Email:
rikuna@moegi.waseda.jp

Received by editor(s):
October 12, 2006

Received by editor(s) in revised form:
January 25, 2007

Published electronically:
December 3, 2007

Article copyright:
© Copyright 2007
American Mathematical Society