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The factorable core of polynomials over finite fields

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

Published online by Cambridge University Press:  09 April 2009

S. D. Cohen
S. D. Cohen
Affiliation:
Department of MathematicsUniversity of Glasgow University GardensGlasgow, G 12, 8QW, Scotland
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Abstract

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For a polynomial f(x) over a finite field Fq, denote the polynomial f(y)−f(x) by ϕf(x, y). The polynomial ϕf has frequently been used in questions on the values of f. The existence is proved here of a polynomial F over Fq of the form F = Lr, where L is an affine linearized polynomial over Fq, such that f = g(F) for some polynomial g and the part of ϕf which splits completely into linear factors over the algebraic closure of Fq is exactly φF. This illuminates an aspect of work of D. R. Hayes and Daqing Wan on the existence of permutation polynomials of even degree. Related results on value sets, including the exhibition of a class of permutation polynomials, are also mentioned.

MSC classification

Secondary: 11T06: Polynomials
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 2 , October 1990 , pp. 309 - 318
Copyright
Copyright © Australian Mathematical Society 1990

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