J Austral Math Soc Ser A 49 pp309--318, 1990.

The Factorable Core of Polynomials over Finite Fields

S. D. Cohen

Abstract

For a polynomial f(x) over a finite field F_q, denote the polynomial f(y) - f(x) by j_f(x, y). The polynomial j_f has frequently been used in questions on the values of f. The existence is proved here of a polynomial F over F_q of the form F = L^r, where L is an affine linearized polynomial over F_q, such that f = g(F) for some polynomial g and the part of j_f which splits completely into linear factors over the algebraic closure of F_q is exactly j_f. 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.

1980 AMS Subject Classification (1985 Revision): 11T06

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Authors

S. D. Cohen

Department of Mathematics, University of Glasgow, University Gardens, Glasgow, G12, 8QW, Scotland.