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On Schur's conjecture

Part of: Finite fields and commutative rings (number-theoretic aspects) General field theory

Published online by Cambridge University Press:  09 April 2009

Gerhard Turnwald
Gerhard Turnwald
Affiliation:
Mathematisches Institut der Universität, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
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Abstract

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We study polynomials over an integral domain R which, for infinitely many prime ideals P, induce a permutation of R/P. In many cases, every polynomial with this property must be a composition of Dickson polynomials and of linear polynomials with coefficients in the quotient field of R. In order to find out which of these compositions have the required property we investigate some number theoretic aspects of composition of polynomials. The paper includes a rather elementary proof of ‘Schur's Conjecture’ and contains a quantitative version for polynomials of prime degree.

MSC classification

Secondary: 11T06: Polynomials 12E05: Polynomials (irreducibility, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 58 , Issue 3 , June 1995 , pp. 312 - 357
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Barton, D. R. and Zippel, R., ‘Polynomial decomposition’, in: Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation (SYMSAC 76; 08 10–12, 1976, Yorktown Heights, New York; Ed.: Jenks, R. D.), pp. 356–358.Google Scholar
[2]Barton, D. R. and Zippel, R., ‘Polynomial decomposition algorithms’, J. Symbolic Comput. 1 (1985), 159–168.CrossRefGoogle Scholar
[3]Birch, B. J. and Swinnerton-Dyer, H. P. F., ‘Note on a problem of Chowla’, Acta Arith. 5 (1959), 417–423.CrossRefGoogle Scholar
[4]Chew, K. L. and Lawn, S., ‘Residually finite rings’, Canad. J. Math. 22 (1970), 92–101.CrossRefGoogle Scholar
[5]Cohen, S. D., ‘The distribution of polynomials over finite fields’, Acta Arith. 17 (1970), 255–271.CrossRefGoogle Scholar
[6]Cohen, S. D., ‘Proof of a conjecture of Chowla and Zassenhaus on permutation polynomials’, Canad. Math. Bull. 33 (1990), 230–234.CrossRefGoogle Scholar
[7]Cohen, S. D., ‘The factorable core of polynomials over finite fields’, J. Austral. Math. Soc. (Series A) 49 (1990), 309–318.CrossRefGoogle Scholar
[8]Cohen, S. D., ‘Exceptional polynomials and the reducibility of substitution polynomials’, Enseign. Math. 36 (1990), 53–65.Google Scholar
[9]Davenport, H. and Lewis, D. J., ‘Notes on congruences I’, Quart. J. Math. Oxford Ser. (2) 14 (1963), 51–60.CrossRefGoogle Scholar
[10]Dickson, L. E., ‘The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group’, Ann. of Math. 11 (1896/1897), 65–120, 161–183.CrossRefGoogle Scholar
[11]Dorey, F. and Whaples, G., ‘Prime and composite polynomials’, J. Algebra 28 (1974), 88–101.CrossRefGoogle Scholar
[12]Evyatar, A. and Scott, D. B., ‘On polynomials in a polynomial’, Bull. London Math. Soc. 4 (1972), 176–178.CrossRefGoogle Scholar
[13]Fried, M., ‘Arithmetical properties of value sets of polynomials’, Acta Arith. 15 (1968/1969), 91–115.CrossRefGoogle Scholar
[14]Fried, M., ‘On a conjecture of Schur’, Michigan Math. J. 17 (1970), 41–55.CrossRefGoogle Scholar
[15]Fried, M., ‘On a theorem of MacCluer’, Acta Arith. 25 (1973/1974), 121–126.CrossRefGoogle Scholar
[16]Fried, M., ‘Arithmetical properties of function fields (II). The generalized Schur problem’, Acta Arith. 25 (1973/1974), 225–258.CrossRefGoogle Scholar
[17]Fried, M., ‘Galois groups and complex multiplication’, Trans. Amer. Math. Soc. 235 (1978), 141–163.CrossRefGoogle Scholar
[18]Fried, M., ‘Exposition on an arithmetic group-theoretic connection via Riemann's existence theorem’, Proc. Sympos. Pure Math. 37 (1980), 571–602.CrossRefGoogle Scholar
[19]Fried, M. and Jarden, M., Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 11 (Springer, New York, 1986).CrossRefGoogle Scholar
[20]Hasse, H., ‘Zwei Existenztheoreme über algebraische Zahlkörper’, Math. Ann. 95 (1926), 229–238.CrossRefGoogle Scholar
[21]Hasse, H., Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz, 2.Auflage (Physica-Verlag, Würzburg, 1965).CrossRefGoogle Scholar
[22]Hering, H., ‘Über Koeffizientenbeschränkungen affektloser Gleichungen’, Math. Ann. 195 (1972), 121–136.CrossRefGoogle Scholar
[23]Hubert, D., ‘Ueber die Irreducibilität ganzer rationaler Functionen mit ganzzahligen Koefficienten’, J. Reine Angew. Math. 110 (1892), 104–129.Google Scholar
(Gesammelte Abhandlungen II (Springer, Berlin, 1970) pp. 264–286.)Google Scholar
[24]Huppert, B., Endliche Gruppen I, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134, (Springer, Berlin, 1967).CrossRefGoogle Scholar
[25]Hurwitz, A., ‘Über einen Fundamentalsatz der arithmetischen Theorie der algebraischen Grössen’, in: Mathematische Werke II (Birkhäuser, Basel, 1933), pp. 198–207.Google Scholar
[26]Janusz, G., Algebraic number fields (Academic Press, New York, 1973).Google Scholar
[27]Kurbatov, V. A., ‘Generalization of Schur's theorem concerning a class of algebraic functions’, Mat. Sb. 21 (63) (1947), 133–140 (in Russian);Google Scholar
Amer. Math. Soc. Transl. Ser. 2 37 (1964), 1–11.Google Scholar
[28]Kurbatov, V. A., ‘On the monodromy group of an algebraic function’, Mat. Sb. 25 (67) (1949), 51–94 (in Russian);Google Scholar
Amer. Math. Soc. Transl. Ser. 2 36 (1964), 17–62.Google Scholar
[29]Lang, S., Algebra, (Addison-Wesley, Reading, 1965).Google Scholar
[30]Lausch, H. and Nöbauer, W., Algebra of polynomials (North-Holland, Amsterdam, 1973).Google Scholar
[31]Lidl, R. and Niederreiter, H., Finite fields, Encyclopedia of Mathematics and its Applications 20 (Addison-Wesley, Reading, 1983) (Reprinted 1987 by Cambridge University Press).Google Scholar
[32]Matthews, R., ‘Permutation polynomials over algebraic number fields’, J. Number Theory 18 (1984), 249–260.CrossRefGoogle Scholar
[33]Matzat, B. H., Konstruktive Galoistheorie, Lecture Notes in Math. 1284 (Springer, Berlin, 1987).CrossRefGoogle Scholar
[34]Narkiewicz, W., Uniform distribution of sequences of integers in residue classes, Lecture Notes in Math. 1087 (Springer, Berlin, 1984).CrossRefGoogle Scholar
[35]Niederreiter, H. and Lo, S. K., ‘Permutation polynomials over rings of algebraic integers’, Abh. Math. Sem. Univ. Hamburg 49 (1979), 126–139.CrossRefGoogle Scholar
[36]Nöbauer, R., ‘Rédei-Funktionen und das Schur'sche Problem’, Arch. Math. 52 (1989), 61–65.CrossRefGoogle Scholar
[37]Nöbauer, W., ‘Polynome, welche für gegebene Zahlen Permutationspolynome sind’, Acta Arith. 11 (1966), 437–442.CrossRefGoogle Scholar
[38]Ore, Ö., ‘Zur Theorie der Eisensteinschen Gleichungen’, Math. Z. 20 (1924), 267–279.CrossRefGoogle Scholar
[39]Passman, D., Permutation groups (Benjamin, New York, 1968).Google Scholar
[40]Ribenboim, P., Algebraic numbers (Wiley, New York, 1972).Google Scholar
[41]Ritt, J. F., ‘Prime and composite polynomials’, Trans. Amer. Math. Soc. 23 (1922), 51–66.CrossRefGoogle Scholar
[42]Ritt, J. F., ‘On algebraic fuctions which can be expressed in terms of radicals’, Trans. Amer. Math. Soc. 24 (1923), 21–30.CrossRefGoogle Scholar
[43]Ritt, J. F., ‘Permutable rational functions’, Trans. Amer. Math. Soc 25 (1923), 399–448.CrossRefGoogle Scholar
[44]Ruppert, W., ‘Reduzibilität ebener Kurven’, J. Reine Angew. Math. 369 (1986), 167–191.Google Scholar
[45]Samuel, P., ‘About Euclidean rings’, J. Algebra 19 (1971), 282–301.CrossRefGoogle Scholar
[46]Schinzel, A., Selected topics on polynomials (University of Michigan Press, Ann Arbor, 1982).CrossRefGoogle Scholar
[47]Schmidt, W. M., Equations over finite fields: An elementary approach, Lecture Notes in Math. 536 (Springer, Berlin, 1976).CrossRefGoogle Scholar
[48]Schur, I., ‘Über den Zusammenhang zwischen einem Problem der Zahlentheorie und einem Satz über algebraische Funktionen’, S. B. Preuss. Akad. Wiss. Berlin (1923), 123–134.Google Scholar
[49]Schur, I., Zur Theorie der einfach transitiven Permutationsgruppen, (S.-B. Preuss. Akad. Wiss. Berlin, 1933), 598–623.Google Scholar
[50]Turnwald, G., ‘On a problem concerning permutation polynomials’, Trans. Amer. Math. Soc. 302 (1987), 251–267.CrossRefGoogle Scholar
[51]Uzkov, A. I., ‘Additional information concerning the content of the product of polynomials’, Math. Notes 16 (1974), 825–827.CrossRefGoogle Scholar
[52]Wegner, U., Über die ganzzahligen Polynome, die für unendlich viele Primzahlmoduln Permutationen liefern (Dissertation, Berlin, 1928).Google Scholar
[53]Wegner, U., ‘Über einen Satz von Dickson’, Math. Ann. 105 (1931), 790–792.CrossRefGoogle Scholar
[54]Wielandt, H., Finite permutation groups (Academic Press, New York, 1964).Google Scholar
[55]Williams, K. S., ‘Note on Dickson's permutation polynomials’, Duke Math. J. 38 (1971), 659–665.CrossRefGoogle Scholar
[56]Cohen, S. D. and Matthews, R. W., ‘A class of exceptional polynomials’, Trans. Amer. Math. Soc. 345 (1994), 897–909.CrossRefGoogle Scholar
[57]Fried, M. D., Guralnick, R. and Saxl, J., ‘Schur covers and Carlitz's conjecture’, Israel J. Math. 82 (1993), 157–225.CrossRefGoogle Scholar
[58]Hayes, D. R., ‘The Galois group of xn + x – t’, Duke Math. J. 40 (1973), 459–461.CrossRefGoogle Scholar
[59]Kljačko, A. A., ‘Monodromy groups of polynomial mappings’, in: Studies in Number Theory, No. 6, (Izdat. Saratov. Univ., Saratov, 1975), pp. 82–91 (in Russian).Google Scholar
[60]Leep, D. B. and Yeomans, C. C., ‘The number of points on a singular curve over a finite field’, Arch. Math. 63 (1994), 420–426.CrossRefGoogle Scholar
[61]Lidl, R., Mullen, G. L., and Turnwald, G., Dickson polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics 65, (Longman, Essex, 1993).Google Scholar
[62]Turnwald, G., ‘A new criterion for permutation polynomials’, Finite Fields Appl. 1 (1995), 64–82.CrossRefGoogle Scholar