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On Schur's conjecture

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

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

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