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Permutation polynomials over finite fields. (English) Zbl 0636.12011
The author proves that if \(q\) is odd and congruent to 1 modulo 3, then the polynomial \(f(x)=x^{1+(q-1)/3}+ax\) \((a\neq 0)\) is not a permutation polynomial over any field \(\mathbb F_{q^r}\) \((r\geq 2)\). Thereby a question raised by L. Carlitz [Bull. Am. Math. Soc. 68, 120–122 (1962); Zbl 0217.33003)] receives a partial answer. In addition, it is shown that if \(p\) is an odd prime, \(1<k<p\), \(p-1>(k-1,p-1)(k-1)\), then \(f(x)=x^k+ax\) \((a\neq 0)\) is not a permutation polynomial over \(\mathbb F_ p\). This result follows from Newton’s identities of power sums as used by L. E. Dickson [Linear groups. 2nd ed. (1958; Zbl 0082.24901), §§ 74, 75 and 84].
Reviewer: H. Lausch

11T06 Polynomials over finite fields
Full Text: DOI
[1] Carlitz, L., Some theorems on permutation polynomials,Bull. Amer. Math. Soc.,68(1962), 120–122. · Zbl 0217.33003 · doi:10.1090/S0002-9904-1962-10750-2
[2] Dickson L. E., Linear Groups,Dover, New York, 1958.
[3] Hayes, D. R., A geometric approach to permutation polynomials over a finite field,Duke Math. J.,34(1967) 293–305. · Zbl 0163.05202 · doi:10.1215/S0012-7094-67-03433-3
[4] Lausch, H. and Nobauer, W., Algebra of Polynomials, North-Holland, Amsterdam, 1973.
[5] Lidl, R., and Niederreiter, H., Finite Fields,Addison-Wesley Publ. Co., Reading, Mass., 1983, Chapter 7. · Zbl 0554.12010
[6] van Lint, L. H., Introduction to Coding Theory,Springer-Verlag, New York, 1982, p. 47. · Zbl 0485.94015
[7] Niederreiter, H. and Robinson, K. H., Complete mappings of finite fields,J. Austral. Math. Soc. Ser. A,33(1982), 197–212. · Zbl 0495.12018 · doi:10.1017/S1446788700018346
[8] Schmidt, W. M., Equations over Finite Fields, Lecture Notes in Math., Springer-Verlage, New York,536(1976). · Zbl 0329.12001
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