Noether’s problem and $\mathbb {Q}$-generic polynomials for the normalizer of the $8$-cycle in $S_8$ and its subgroups
HTML articles powered by AMS MathViewer
- by Ki-ichiro Hashimoto, Akinari Hoshi and Yuichi Rikuna PDF
- Math. Comp. 77 (2008), 1153-1183 Request permission
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.References
- J. Buhler and Z. Reichstein, On the essential dimension of a finite group, Compositio Math. 106 (1997), no. 2, 159–179. MR 1457337, DOI 10.1023/A:1000144403695
- 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 10.1007/s00014-003-0783-8
- Frank R. DeMeyer, Generic polynomials, J. Algebra 84 (1983), no. 2, 441–448. MR 723401, DOI 10.1016/0021-8693(83)90087-X
- Frank DeMeyer and Thomas McKenzie, On generic polynomials, J. Algebra 261 (2003), no. 2, 327–333. MR 1966633, DOI 10.1016/S0021-8693(02)00678-6
- Shizuo Endô and Takehiko Miyata, Invariants of finite abelian groups, J. Math. Soc. Japan 25 (1973), 7–26. MR 311754, DOI 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, DOI 10.1090/ulect/028
- Mowaffaq Hajja, A note on monomial automorphisms, J. Algebra 85 (1983), no. 1, 243–250. MR 723077, DOI 10.1016/0021-8693(83)90128-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 10.1016/0021-8693(87)90162-1
- Mowaffaq Hajja and Ming Chang Kang, Three-dimensional purely monomial group actions, J. Algebra 170 (1994), no. 3, 805–860. MR 1305266, DOI 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 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 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 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 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 10.1007/BF02568311
- Gregor Kemper, Generic polynomials are descent-generic, Manuscripta Math. 105 (2001), no. 1, 139–141. MR 1885819, DOI 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 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 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 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 10.1007/BF01389732
- Gunter Malle and B. Heinrich Matzat, Inverse Galois theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. MR 1711577, DOI 10.1007/978-3-662-12123-8
- 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 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 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 10.1016/0001-8708(82)90036-6
- David J. Saltman, Retract rational fields and cyclic Galois extensions, Israel J. Math. 47 (1984), no. 2-3, 165–215. MR 738167, DOI 10.1007/BF02760515
- Leila Schneps, On cyclic field extensions of degree $8$, Math. Scand. 71 (1992), no. 1, 24–30. MR 1216101, DOI 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 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 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 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, DOI 10.1090/mmono/179
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
- © Copyright 2007 American Mathematical Society
- Journal: Math. Comp. 77 (2008), 1153-1183
- MSC (2000): Primary 12F12, 14E08, 11R32
- DOI: https://doi.org/10.1090/S0025-5718-07-02094-7
- MathSciNet review: 2373196