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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

MSC:
11T06 Polynomials over finite fields
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