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Permutation binomials over finite fields. (English) Zbl 0817.11056
For \(q\) a prime power let \(\mathbb{F}_ q\) denote the finite field of order \(q\). A polynomial \(f\in \mathbb{F}_ q [x]\) is a permutation polynomial (PP) if \(f\) induces a 1-1 mapping of \(\mathbb{F}_ q\) onto itself. In a previous paper [Acta Math. Sin., New Ser. 3, 1-5 (1987; Zbl 0636.12011)], the author proved that for odd \(q\), the polynomial \(x^{1+ (q-1) /3}+ ax\) with \(a\neq 0\in \mathbb{F}_ q\) is not a PP over \(\mathbb{F}_{q^ r}\) with \(r\geq 2\). In the present paper the author proves that the same conclusion holds in the case when \(q\) is even. For related results see the author and R. Lidl [Monatsh. Math. 112, 149-163 (1991; Zbl 0737.11040)].

MSC:
11T06 Polynomials over finite fields
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