Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T07:54:55.807Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Agou, S., ‘Sur I'irréductibilité des trinòmes sur les corps finis ’, Acta Arith. 44 (1984), 343–355.CrossRefGoogle Scholar
[2]Agou, S., ‘Sur la factorisation des polynòmes sur les corps finis ’, Manuscripta Math. 58 (1987), 141–154.CrossRefGoogle Scholar
[3]Carlitz, L., ‘On factorable polynomials in several indeterminates’, Duke Math. J. 2 (1936), 660–670.CrossRefGoogle Scholar
[4]Carlitz, L., ‘On the number of distinct values of a polynomial with coefficients in a finite field’, Proc. Japan Acad. 31 (1955), 119–120.Google Scholar
[5]Cohen, S. D., ‘The distribution of polynomials over finite fields’, Acta Arith. 17 (1970), 255–271.CrossRefGoogle Scholar
[6]Cohen, S. D., ‘Uniform distribution of polynomials over finite fields’, J. London Math. Soc. (2) 6 (1972), 93–102.CrossRefGoogle Scholar
[7]Cohen, S. D., ‘The irreducibility of compositions of linear polynomials over a finite field’, Compositio Math. 47 (1982), 149–152.Google Scholar
[8]Cohen, S. D., ‘Reducibility of sub-linear polynomials over a finite field’, Bull. Korean Math. Soc. 22 (1985), 53–56.Google Scholar
[9]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 (1897), 65–120 and 161–183.CrossRefGoogle Scholar
[10]Fried, M. D. and MacRae, R. E., ‘On curves with separated variables’, Math. Ann. 180 (1969), 220–226.CrossRefGoogle Scholar
[11]Gomez-Calderon, J. and Madden, D. J., ‘Polynomials with small value set over finite fields’, J. Number Theory 28 (1988), 167–188.CrossRefGoogle Scholar
[12]Gomez-Calderon, J., ‘A note on polynomials with minimal value set over finite fields’, Mathematika 35 (1988), 144–148.CrossRefGoogle Scholar
[13]Hayes, D. R., ‘A geometric approach to permutation polynomials over a finite field’, Duke Math. J. 34 (1967), 293–305.CrossRefGoogle Scholar
[14]Lidl, R. and Mullen, G. L., ‘When does a polynomial over a finite field permute the elements of the field?’, Amer. Math. Monthly 95 (1988), 243–246.CrossRefGoogle Scholar
[15]Lidl, R. and Niederreiter, H., Finite Fields, (Addision-Wesley, Reading, Massachusetts, 1983). (Now distributed by Cambridge University Press, Cambridge.)Google Scholar
[16]Long, A. F., ‘Some theorems on factorable irreducible polynomials’, Duke Math. J. 34 (1967), 281–291.CrossRefGoogle Scholar
[17]Mills, W. H., ‘Polynomials with minimal value sets’, Pacific J. Math. 14 (1964), 225–241.CrossRefGoogle Scholar
[18]Mills, W. H., ‘The degree of factors of certain polynomials over finite fields’, Proc. Amer. Math. Soc. 25 (1970), 860–863.Google Scholar
[19]Mit' kin, D. A., ‘Polynomials with a minimal value set of values and the equation f(x) = f(y) in a finite prime field’, (Russian), Mat. Zametki 38 (1985), 3–14, 168.Google Scholar
[20]Wan, Daqing, ‘On a conjecture of Carlitz’, J. Austral. Math. Soc. Ser. A 43 (1987), 375–384.Google Scholar