Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Kronecker conjugacy of polynomials


Author: Peter Müller
Journal: Trans. Amer. Math. Soc. 350 (1998), 1823-1850
MSC (1991): Primary 11C08, 20B10; Secondary 11R09, 12E05, 12F10, 20B20, 20D05
MathSciNet review: 1458331
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $f,g\in \mathbb{Z}[X]$ be non-constant polynomials with integral coefficients. In 1968 H. Davenport raised the question as to when the value sets $f(\mathbb{Z})$ and $g(\mathbb{Z})$ are the same modulo all but finitely many primes. The main progress until now is M. Fried's result that $f$ and $g$ then differ by a linear substitution, provided that $f$ is functionally indecomposable. We extend this result to polynomials $f$ of composition length $2$. Also, we study the analog when $\mathbb{Z}$ is replaced by the integers of a number field. The above number theoretic property translates to an equivalent property of subgroups of a finite group, known as Kronecker conjugacy, a weakening of conjugacy which has been studied by various authors under different assumptions and in other contexts.

We also give a simplified and strengthened version of the Galois theoretic translation to finite groups.


References [Enhancements On Off] (What's this?)

  • 1. Salomon Bochner, The theorem of Morera in several variables, Ann. Mat. Pura Appl. (4) 34 (1953), 27–39. MR 0057341
  • 2. Gregory Butler and John McKay, The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), no. 8, 863–911. MR 695893, 10.1080/00927878308822884
  • 3. Claude Chevalley, Introduction to the Theory of Algebraic Functions of One Variable, Mathematical Surveys, No. VI, American Mathematical Society, New York, N. Y., 1951. MR 0042164
  • 4. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
  • 5. Walter Feit, On symmetric balanced incomplete block designs with doubly transitive automorphism groups, J. Combinatorial Theory Ser. A 14 (1973), 221–247. MR 0327540
  • 6. Walter Feit, Some consequences of the classification of finite simple groups, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979) Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 175–181. MR 604576
  • 7. Michael Fried, On a conjecture of Schur, Michigan Math. J. 17 (1970), 41–55. MR 0257033
  • 8. Michael Fried, The field of definition of function fields and a problem in the reducibility of polynomials in two variables, Illinois J. Math. 17 (1973), 128–146. MR 0347828
  • 9. Michael Fried, On Hilbert’s irreducibility theorem, J. Number Theory 6 (1974), 211–231. MR 0349624
  • 10. M. Fried, Rigidity and applications of the classification of simple groups to monodromy, Part II - Applications of connectivity; Davenport and Hilbert-Siegel Problems, preprint.
  • 11. Michael D. Fried, Extension of constants, rigidity, and the Chowla-Zassenhaus conjecture, Finite Fields Appl. 1 (1995), no. 3, 326–359. MR 1341951, 10.1006/ffta.1995.1025
  • 12. Michael D. Fried and Moshe Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 11, Springer-Verlag, Berlin, 1986. MR 868860
  • 13. Michael D. Fried and R. E. MacRae, On the invariance of chains of fields, Illinois J. Math. 13 (1969), 165–171. MR 0238815
  • 14. Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
  • 15. Robert M. Guralnick, Zeroes of permutation characters with applications to prime splitting and Brauer groups, J. Algebra 131 (1990), no. 1, 294–302. MR 1055010, 10.1016/0021-8693(90)90177-P
  • 16. Robert M. Guralnick, Subgroups inducing the same permutation representation, J. Algebra 81 (1983), no. 2, 312–319. MR 700287, 10.1016/0021-8693(83)90191-6
  • 17. Wolfram Jehne, Kronecker classes of algebraic number fields, J. Number Theory 9 (1977), no. 3, 279–320. MR 0447184
  • 18. Norbert Klingen, Zahlkörper mit gleicher Primzerlegung, J. Reine Angew. Math. 299/300 (1978), 342–384 (German). MR 0491594
  • 19. L. Kronecker, Über die Irreduzibilität von Gleichungen, Werke II, 85-93; Monatsberichte Deutsche Akademie für Wissenschaft (1880), 155-163.
  • 20. B. Heinrich Matzat, Konstruktion von Zahl- und Funktionenkörpern mit vorgegebener Galoisgruppe, J. Reine Angew. Math. 349 (1984), 179–220 (German). MR 743971, 10.1515/crll.1984.349.179
  • 21. Peter Müller, Primitive monodromy groups of polynomials, Recent developments in the inverse Galois problem (Seattle, WA, 1993), Contemp. Math., vol. 186, Amer. Math. Soc., Providence, RI, 1995, pp. 385–401. MR 1352284, 10.1090/conm/186/02193
  • 22. P. Müller, Reducibility behavior of polynomials with varying coefficients, Israel J. Math. 94 (1996), 59-91. CMP 96:14
  • 23. P. Müller, An infinite series of Kronecker conjugate polynomials, Proc. Amer. Math. Soc. 125 (1997), 1933-1940. CMP 97:10
  • 24. Peter Müller and Helmut Völklein, On a question of Davenport, J. Number Theory 58 (1996), no. 1, 46–54. MR 1387720, 10.1006/jnth.1996.0059
  • 25. Robert Perlis, On the equation 𝜁_{𝐾}(𝑠)=𝜁_{𝐾’}(𝑠), J. Number Theory 9 (1977), no. 3, 342–360. MR 0447188
  • 26. Cheryl E. Praeger, Kronecker classes of field extensions of small degree, J. Austral. Math. Soc. Ser. A 50 (1991), no. 2, 297–315. MR 1094925
  • 27. J. F. Ritt, Prime and composite polynomials, Trans. Amer. Math. Soc. 23 (1922), 51-66.
  • 28. Gordon F. Royle, The transitive groups of degree twelve, J. Symbolic Comput. 4 (1987), no. 2, 255–268. MR 922391, 10.1016/S0747-7171(87)80068-8
  • 29. Jan Saxl, On a question of W. Jehne concerning covering subgroups of groups and Kronecker classes of fields, J. London Math. Soc. (2) 38 (1988), no. 2, 243–249. MR 966296, 10.1112/jlms/s2-38.2.243
  • 30. M. Schönert et. al., GAP - Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, Rheinisch-Westfälische Techn. Hochschule, Aachen, Germany, fourth edition, 1994.
  • 31. I. Schur, Über den Zusammenhang zwischen einem Problem der Zahlentheorie und einem Satz über algebraische Funktionen, S.-B. Preuss. Akad. Wiss., Phys.-Math. Klasse (1923), 123-134.
  • 32. E. Trost, Zur Theorie der Potenzreste, Nieuw Arch.Wiskunde 18 (1934), 58-61.
  • 33. Gerhard Turnwald, On Schur’s conjecture, J. Austral. Math. Soc. Ser. A 58 (1995), no. 3, 312–357. MR 1329867
  • 34. H. Völklein. Groups as Galois Groups - an Introduction, Cambridge University Press, 1996. CMP 96:17
  • 35. Helmut Wielandt, Finite permutation groups, Translated from the German by R. Bercov, Academic Press, New York-London, 1964. MR 0183775

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11C08, 20B10, 11R09, 12E05, 12F10, 20B20, 20D05

Retrieve articles in all journals with MSC (1991): 11C08, 20B10, 11R09, 12E05, 12F10, 20B20, 20D05


Additional Information

Peter Müller
Affiliation: IWR, Universität Heidelberg, D-69120 Heidelberg, Germany
Email: peter.mueller@iwr.uni-heidelberg.de

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02123-0
Received by editor(s): January 16, 1996
Additional Notes: The author thanks the Deutsche Forschungsgemeinschaft (DFG) for its support in the form of a postdoctoral fellowship
Article copyright: © Copyright 1998 American Mathematical Society