Kronecker conjugacy of polynomials

Müller, Peter

doi:10.1090/S0002-9947-98-02123-0

Kronecker conjugacy of polynomials
HTML articles powered by AMS MathViewer

by Peter Müller

Trans. Amer. Math. Soc. 350 (1998), 1823-1850

DOI: https://doi.org/10.1090/S0002-9947-98-02123-0

PDF | Request permission

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

Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
Gregory Butler and John McKay, The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), no. 8, 863–911. MR 695893, DOI 10.1080/00927878308822884
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ 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
Walter Feit, On symmetric balanced incomplete block designs with doubly transitive automorphism groups, J. Combinatorial Theory Ser. A 14 (1973), 221–247. MR 327540, DOI 10.1016/0097-3165(73)90024-1
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
Michael Fried, On a conjecture of Schur, Michigan Math. J. 17 (1970), 41–55. MR 257033
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 347828
Michael Fried, On Hilbert’s irreducibility theorem, J. Number Theory 6 (1974), 211–231. MR 349624, DOI 10.1016/0022-314X(74)90015-8
M. Fried, Rigidity and applications of the classification of simple groups to monodromy, Part II – Applications of connectivity; Davenport and Hilbert-Siegel Problems, preprint.
Michael D. Fried, Extension of constants, rigidity, and the Chowla-Zassenhaus conjecture, Finite Fields Appl. 1 (1995), no. 3, 326–359. MR 1341951, DOI 10.1006/ffta.1995.1025
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, DOI 10.1007/978-3-662-07216-5
Michael D. Fried and R. E. MacRae, On the invariance of chains of fields, Illinois J. Math. 13 (1969), 165–171. MR 238815
Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
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, DOI 10.1016/0021-8693(90)90177-P
Robert M. Guralnick, Subgroups inducing the same permutation representation, J. Algebra 81 (1983), no. 2, 312–319. MR 700287, DOI 10.1016/0021-8693(83)90191-6
Wolfram Jehne, Kronecker classes of algebraic number fields, J. Number Theory 9 (1977), no. 3, 279–320. MR 447184, DOI 10.1016/0022-314X(77)90066-X
Norbert Klingen, Zahlkörper mit gleicher Primzerlegung, J. Reine Angew. Math. 299(300) (1978), 342–384 (German). MR 491594, DOI 10.1515/crll.1978.299-300.342
L. Kronecker, Über die Irreduzibilität von Gleichungen, Werke II, 85–93; Monatsberichte Deutsche Akademie für Wissenschaft (1880), 155–163.
B. Heinrich Matzat, Konstruktion von Zahl- und Funktionenkörpern mit vorgegebener Galoisgruppe, J. Reine Angew. Math. 349 (1984), 179–220 (German). MR 743971, DOI 10.1515/crll.1984.349.179
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, DOI 10.1090/conm/186/02193
P. Müller, Reducibility behavior of polynomials with varying coefficients, Israel J. Math. 94 (1996), 59–91.
P. Müller, An infinite series of Kronecker conjugate polynomials, Proc. Amer. Math. Soc. 125 (1997), 1933–1940.
Peter Müller and Helmut Völklein, On a question of Davenport, J. Number Theory 58 (1996), no. 1, 46–54. MR 1387720, DOI 10.1006/jnth.1996.0059
Robert Perlis, On the equation $\zeta _{K}(s)=\zeta _{K’}(s)$, J. Number Theory 9 (1977), no. 3, 342–360. MR 447188, DOI 10.1016/0022-314X(77)90070-1
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
J. F. Ritt, Prime and composite polynomials, Trans. Amer. Math. Soc. 23 (1922), 51–66.
Gordon F. Royle, The transitive groups of degree twelve, J. Symbolic Comput. 4 (1987), no. 2, 255–268. MR 922391, DOI 10.1016/S0747-7171(87)80068-8
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, DOI 10.1112/jlms/s2-38.2.243
M. Schönert et. al., GAP – Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, Rheinisch-Westfälische Techn. Hochschule, Aachen, Germany, fourth edition, 1994.
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.
E. Trost, Zur Theorie der Potenzreste, Nieuw Arch. Wiskunde 18 (1934), 58–61.
Gerhard Turnwald, On Schur’s conjecture, J. Austral. Math. Soc. Ser. A 58 (1995), no. 3, 312–357. MR 1329867
H. Völklein. Groups as Galois Groups – an Introduction, Cambridge University Press, 1996.
Helmut Wielandt, Finite permutation groups, Academic Press, New York-London, 1964. Translated from the German by R. Bercov. MR 0183775