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On Schur's conjecture

  • Gerhard Turnwald (a1)

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.

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[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. 356358.
[2]Barton, D. R. and Zippel, R., ‘Polynomial decomposition algorithms’, J. Symbolic Comput. 1 (1985), 159168.
[3]Birch, B. J. and Swinnerton-Dyer, H. P. F., ‘Note on a problem of Chowla’, Acta Arith. 5 (1959), 417423.
[4]Chew, K. L. and Lawn, S., ‘Residually finite rings’, Canad. J. Math. 22 (1970), 92101.
[5]Cohen, S. D., ‘The distribution of polynomials over finite fields’, Acta Arith. 17 (1970), 255271.
[6]Cohen, S. D., ‘Proof of a conjecture of Chowla and Zassenhaus on permutation polynomials’, Canad. Math. Bull. 33 (1990), 230234.
[7]Cohen, S. D., ‘The factorable core of polynomials over finite fields’, J. Austral. Math. Soc. (Series A) 49 (1990), 309318.
[8]Cohen, S. D., ‘Exceptional polynomials and the reducibility of substitution polynomials’, Enseign. Math. 36 (1990), 5365.
[9]Davenport, H. and Lewis, D. J., ‘Notes on congruences I’, Quart. J. Math. Oxford Ser. (2) 14 (1963), 5160.
[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), 65120, 161–183.
[11]Dorey, F. and Whaples, G., ‘Prime and composite polynomials’, J. Algebra 28 (1974), 88101.
[12]Evyatar, A. and Scott, D. B., ‘On polynomials in a polynomial’, Bull. London Math. Soc. 4 (1972), 176178.
[13]Fried, M., ‘Arithmetical properties of value sets of polynomials’, Acta Arith. 15 (1968/1969), 91115.
[14]Fried, M., ‘On a conjecture of Schur’, Michigan Math. J. 17 (1970), 4155.
[15]Fried, M., ‘On a theorem of MacCluer’, Acta Arith. 25 (1973/1974), 121126.
[16]Fried, M., ‘Arithmetical properties of function fields (II). The generalized Schur problem’, Acta Arith. 25 (1973/1974), 225258.
[17]Fried, M., ‘Galois groups and complex multiplication’, Trans. Amer. Math. Soc. 235 (1978), 141163.
[18]Fried, M., ‘Exposition on an arithmetic group-theoretic connection via Riemann's existence theorem’, Proc. Sympos. Pure Math. 37 (1980), 571602.
[19]Fried, M. and Jarden, M., Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 11 (Springer, New York, 1986).
[20]Hasse, H., ‘Zwei Existenztheoreme über algebraische Zahlkörper’, Math. Ann. 95 (1926), 229238.
[21]Hasse, H., Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz, 2.Auflage (Physica-Verlag, Würzburg, 1965).
[22]Hering, H., ‘Über Koeffizientenbeschränkungen affektloser Gleichungen’, Math. Ann. 195 (1972), 121136.
[23]Hubert, D., ‘Ueber die Irreducibilität ganzer rationaler Functionen mit ganzzahligen Koefficienten’, J. Reine Angew. Math. 110 (1892), 104129.
(Gesammelte Abhandlungen II (Springer, Berlin, 1970) pp. 264286.)
[24]Huppert, B., Endliche Gruppen I, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134, (Springer, Berlin, 1967).
[25]Hurwitz, A., ‘Über einen Fundamentalsatz der arithmetischen Theorie der algebraischen Grössen’, in: Mathematische Werke II (Birkhäuser, Basel, 1933), pp. 198207.
[26]Janusz, G., Algebraic number fields (Academic Press, New York, 1973).
[27]Kurbatov, V. A., ‘Generalization of Schur's theorem concerning a class of algebraic functions’, Mat. Sb. 21 (63) (1947), 133140 (in Russian);
Amer. Math. Soc. Transl. Ser. 2 37 (1964), 111.
[28]Kurbatov, V. A., ‘On the monodromy group of an algebraic function’, Mat. Sb. 25 (67) (1949), 5194 (in Russian);
Amer. Math. Soc. Transl. Ser. 2 36 (1964), 1762.
[29]Lang, S., Algebra, (Addison-Wesley, Reading, 1965).
[30]Lausch, H. and Nöbauer, W., Algebra of polynomials (North-Holland, Amsterdam, 1973).
[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).
[32]Matthews, R., ‘Permutation polynomials over algebraic number fields’, J. Number Theory 18 (1984), 249260.
[33]Matzat, B. H., Konstruktive Galoistheorie, Lecture Notes in Math. 1284 (Springer, Berlin, 1987).
[34]Narkiewicz, W., Uniform distribution of sequences of integers in residue classes, Lecture Notes in Math. 1087 (Springer, Berlin, 1984).
[35]Niederreiter, H. and Lo, S. K., ‘Permutation polynomials over rings of algebraic integers’, Abh. Math. Sem. Univ. Hamburg 49 (1979), 126139.
[36]Nöbauer, R., ‘Rédei-Funktionen und das Schur'sche Problem’, Arch. Math. 52 (1989), 6165.
[37]Nöbauer, W., ‘Polynome, welche für gegebene Zahlen Permutationspolynome sind’, Acta Arith. 11 (1966), 437442.
[38]Ore, Ö., ‘Zur Theorie der Eisensteinschen Gleichungen’, Math. Z. 20 (1924), 267279.
[39]Passman, D., Permutation groups (Benjamin, New York, 1968).
[40]Ribenboim, P., Algebraic numbers (Wiley, New York, 1972).
[41]Ritt, J. F., ‘Prime and composite polynomials’, Trans. Amer. Math. Soc. 23 (1922), 5166.
[42]Ritt, J. F., ‘On algebraic fuctions which can be expressed in terms of radicals’, Trans. Amer. Math. Soc. 24 (1923), 2130.
[43]Ritt, J. F., ‘Permutable rational functions’, Trans. Amer. Math. Soc 25 (1923), 399448.
[44]Ruppert, W., ‘Reduzibilität ebener Kurven’, J. Reine Angew. Math. 369 (1986), 167191.
[45]Samuel, P., ‘About Euclidean rings’, J. Algebra 19 (1971), 282301.
[46]Schinzel, A., Selected topics on polynomials (University of Michigan Press, Ann Arbor, 1982).
[47]Schmidt, W. M., Equations over finite fields: An elementary approach, Lecture Notes in Math. 536 (Springer, Berlin, 1976).
[48]Schur, I., ‘Über den Zusammenhang zwischen einem Problem der Zahlentheorie und einem Satz über algebraische Funktionen’, S. B. Preuss. Akad. Wiss. Berlin (1923), 123134.
[49]Schur, I., Zur Theorie der einfach transitiven Permutationsgruppen, (S.-B. Preuss. Akad. Wiss. Berlin, 1933), 598623.
[50]Turnwald, G., ‘On a problem concerning permutation polynomials’, Trans. Amer. Math. Soc. 302 (1987), 251267.
[51]Uzkov, A. I., ‘Additional information concerning the content of the product of polynomials’, Math. Notes 16 (1974), 825827.
[52]Wegner, U., Über die ganzzahligen Polynome, die für unendlich viele Primzahlmoduln Permutationen liefern (Dissertation, Berlin, 1928).
[53]Wegner, U., ‘Über einen Satz von Dickson’, Math. Ann. 105 (1931), 790792.
[54]Wielandt, H., Finite permutation groups (Academic Press, New York, 1964).
[55]Williams, K. S., ‘Note on Dickson's permutation polynomials’, Duke Math. J. 38 (1971), 659665.
[56]Cohen, S. D. and Matthews, R. W., ‘A class of exceptional polynomials’, Trans. Amer. Math. Soc. 345 (1994), 897909.
[57]Fried, M. D., Guralnick, R. and Saxl, J., ‘Schur covers and Carlitz's conjecture’, Israel J. Math. 82 (1993), 157225.
[58]Hayes, D. R., ‘The Galois group of xn + x – t’, Duke Math. J. 40 (1973), 459461.
[59]Kljačko, A. A., ‘Monodromy groups of polynomial mappings’, in: Studies in Number Theory, No. 6, (Izdat. Saratov. Univ., Saratov, 1975), pp. 8291 (in Russian).
[60]Leep, D. B. and Yeomans, C. C., ‘The number of points on a singular curve over a finite field’, Arch. Math. 63 (1994), 420426.
[61]Lidl, R., Mullen, G. L., and Turnwald, G., Dickson polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics 65, (Longman, Essex, 1993).
[62]Turnwald, G., ‘A new criterion for permutation polynomials’, Finite Fields Appl. 1 (1995), 6482.
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Journal of the Australian Mathematical Society
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