Proceedings of the Japan Academy, Series A, Mathematical Sciences

On Noether’s problem for cyclic groups of prime order

Akinari Hoshi

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Abstract

Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_{g}\mid g\in G)$ by $k$-automorphisms $h(x_{g})=x_{hg}$ for any $g,h\in G$. Noether’s problem asks whether the invariant field $k(G)=k(x_{g}\mid g\in G)^{G}$ is rational (i.e. purely transcendental) over $k$. In 1974, Lenstra gave a necessary and sufficient condition to this problem for abelian groups $G$. However, even for the cyclic group $C_{p}$ of prime order $p$, it is unknown whether there exist infinitely many primes $p$ such that $\mathbf{Q}(C_{p})$ is rational over $\mathbf{Q}$. Only known 17 primes $p$ for which $\mathbf{Q}(C_{p})$ is rational over $\mathbf{Q}$ are $p\leq 43$ and $p=61,67,71$. We show that for primes $p< 20000$, $\mathbf{Q}(C_{p})$ is not (stably) rational over $\mathbf{Q}$ except for affirmative 17 primes and undetermined 46 primes. Under the GRH, the generalized Riemann hypothesis, we also confirm that $\mathbf{Q}(C_{p})$ is not (stably) rational over $\mathbf{Q}$ for undetermined 28 primes $p$ out of 46.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 3 (2015), 39-44.

Dates
First available in Project Euclid: 3 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.pja/1425396669

Digital Object Identifier
doi:10.3792/pjaa.91.39

Mathematical Reviews number (MathSciNet)
MR3317750

Zentralblatt MATH identifier
1334.12007

Subjects
Primary: 11R18: Cyclotomic extensions 11R29: Class numbers, class groups, discriminants 12F12: Inverse Galois theory 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24] 14E08: Rationality questions [See also 14M20] 14F22: Brauer groups of schemes [See also 12G05, 16K50]

Keywords
Noether’s problem rationality problem algebraic tori class number cyclotomic field

Citation

Hoshi, Akinari. On Noether’s problem for cyclic groups of prime order. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 3, 39--44. doi:10.3792/pjaa.91.39. https://projecteuclid.org/euclid.pja/1425396669


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References

  • F. A. Bogomolov, The Brauer group of quotient spaces of linear representations, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 3, 485–516, 688; translation in Math. USSR-Izv. 30 (1988), no. 3, 455–485.
  • F. A. Bogomolov and C. Böhning, Isoclinism and stable cohomology of wreath products, in Birational geometry, rational curves, and arithmetic, Springer, New York, 2013, pp. 57–76.
  • H. Chu, S.-J. Hu, M. Kang and B. E. Kunyavskii, Noether's problem and the unramified Brauer group for groups of order 64, Int. Math. Res. Not. IMRN 2010, no. 12, 2329–2366.
  • S. Endo and T. Miyata, Invariants of finite abelian groups, J. Math. Soc. Japan 25 (1973), 7–26.
  • E. Fischer, Die Isomorphie der Invariantenkörper der endlichen Abel'schen Gruppen linearer Transformationen, Nachr. Königl. Ges. Wiss. Göttingen (1915), 77–80.
  • T. Fukuda and K. Komatsu, Weber's class number problem in the cyclotomic $\mathbf{Z}_{2}$-extension of $\mathbf{Q}$, Experiment. Math. 18 (2009), no. 2, 213–222.
  • T. Fukuda and K. Komatsu, Weber's class number problem in the cyclotomic $\mathbf{Z}_{2}$-extension of $\mathbf{Q}$, II, J. Théor. Nombres Bordeaux 22 (2010), no. 2, 359–368.
  • T. Fukuda and K. Komatsu, Weber's class number problem in the cyclotomic $\mathbf{Z}_{2}$-extension of $\mathbf{Q}$, III, Int. J. Number Theory 7 (2011), no. 6, 1627–1635.
  • A. Hoshi, Multiplicative quadratic forms on algebraic varieties and Noether's problem for meta-abelian groups, Ph. D. dissertation, Waseda University, 2005. http://dspace.wul.waseda.ac.jp/dspace/handle/2065/3004
  • A. Hoshi, On Noether's problem for cyclic groups of prime order, arXiv:1402.3678v2.
  • A. Hoshi, M. Kang and B. E. Kunyavskii, Noether's problem and unramified Brauer groups, Asian J. Math. 17 (2013), no. 4, 689–713.
  • M. Kang, Retract rational fields, J. Algebra 349 (2012), 22–37.
  • M. Kang, Frobenius groups and retract rationality, Adv. Math. 245 (2013), 34–51.
  • M. Kang, Bogomolov multipliers and retract rationality for semidirect products, J. Algebra 397 (2014), 407–425.
  • M. Kang and B. Plans, Reduction theorems for Noether's problem, Proc. Amer. Math. Soc. 137 (2009), no. 6, 1867–1874.
  • H. Kuniyoshi, On purely-transcendency of a certain field, Tohoku Math. J. (2) 6 (1954), 101–108.
  • H. Kuniyoshi, On a problem of Chevalley, Nagoya Math. J. 8 (1955), 65–67.
  • H. Kuniyoshi, Certain subfields of rational function fields, in Proceedings of the international symposium on algebraic number theory (Tokyo & Nikko, 1955), 241–243, Science Council of Japan, Tokyo, 1956.
  • H. W. Lenstra, Jr., Rational functions invariant under a finite abelian group, Invent. Math. 25 (1974), 299–325.
  • H. W. Lenstra, Jr., Rational functions invariant under a cyclic group, in Proceedings of the Queen's Number Theory Conference (Kingston, Ont., 1979), 91–99, Queen's Papers in Pure and Appl. Math., 54, Queen's Univ., Kingston, ON, 1980.
  • J. M. Masley and H. L. Montgomery, Cyclotomic fields with unique factorization, J. Reine Angew. Math. 286/287 (1976), 248–256.
  • K. Masuda, On a problem of Chevalley, Nagoya Math. J. 8 (1955), 59–63.
  • K. Masuda, Application of the theory of the group of classes of projective modules to the existance problem of independent parameters of invariant, J. Math. Soc. Japan 20 (1968), 223–232.
  • J. C. Miller, Class numbers of totally real fields and applications to the Weber class number problem, Acta Arith. 164 (2014), no. 4, 381–398.
  • J. C. Miller, Real cyclotomic fields of prime conductor and their class numbers, arXiv:1407.2373. (to appear in Math. Comp.).
  • P. Moravec, Unramified Brauer groups of finite and infinite groups, Amer. J. Math. 134 (2012), no. 6, 1679–1704.
  • E. Noether, Rationale Funktionenkörper, Jahresber. Deutsch. Math.-Verein. 22 (1913) 316–319.
  • E. Noether, Gleichungen mit vorgeschriebener Gruppe, Math. Ann. 78 (1917), no. 1, 221–229.
  • PARI/GP, version 2.6.0 (alpha), Bordeaux, 2013, http://pari.math.u-bordeaux.fr/.
  • R. G. Swan, Invariant rational functions and a problem of Steenrod, Invent. Math. 7 (1969), 148–158.
  • R. G. Swan, Galois theory, in Emmy Noether. A tribute to her life and work, edited by James W. Brewer and Martha K. Smith, Monographs and Textbooks in Pure and Applied Mathematics, 69, Dekker, New York, 1981.
  • R. G. Swan, Noether's problem in Galois theory, in Emmy Noether in Bryn Mawr (Bryn Mawr, Pa., 1982), edited by B. Srinivasan and J. Sally, 21–40, Springer, New York, 1983.
  • D. J. Saltman, Noether's problem over an algebraically closed field, Invent. Math. 77 (1984), no. 1, 71–84.
  • V. E. Voskresenskiĭ, On the question of the structure of the subfield of invariants of a cyclic group of automorphisms of the field $Q(x_{1},\cdots,\,x_{n})$, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 366–375. English translation: Math. USSR-Izv. 4 (1970), no. 2, 371–380.
  • V. E. Voskresenskiĭ, Rationality of certain algebraic tori, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1037–1046. English translation: Math. USSR-Izv. 5 (1971), no. 5, 1049–1056.
  • V. E. Voskresenskiĭ, Fields of invariants of abelian groups, Uspekhi Mat. Nauk 28 (1973), no. 4 (172), 77–102. English translation: Russian Math. Surveys 28 (1973), no. 4, 79–105.
  • V. E. Voskresenskiĭ, Algebraic groups and their birational invariants, translated from the Russian manuscript by Boris Kunyavski [Boris È. Kunyavskiĭ], Translations of Mathematical Monographs, 179, Amer. Math. Soc., Providence, RI, 1998.
  • L. C. Washington, Introduction to cyclotomic fields, second edition, Graduate Texts in Mathematics, 83, Springer, New York, 1997.