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Binomial permutations of finite fields. (English) Zbl 0648.12013
Let \(F_ q\) denote the finite field of order q. A polynomial \(f\in F_ q[x]\) is a permutation polynomial (PP) if the mapping \(\alpha\to f(\alpha)\), \(\alpha \in F_ q\), is a permutation of \(F_ q\). The question of determining which polynomials over \(F_ q\) permute \(F_ q\) is an interesting, but difficult one. S. R. Cavior [Math. Comput. 17, 450-452 (1963; Zbl 0141.033)] considered binomials of the form x \(8+ax\) j, \(j=1,3,5,7\) and tried to determine when such binomials permute \(F_ q\) with q odd. He asked: Is x \(8+ax\) 5 a PP over \(F_{7\quad n}\) for n odd? Or over \(F_{13\quad n}\) for n odd? Is x \(8+ax\) 3 a PP over \(F_{11\quad n}\) for n odd? Using Hermite’s criterion, the author shows that if \(q=7\) n, x \(8+ax\) 5 permutes \(F_ q\) if and only if \(n=1\) and \(a=3\) or 4; if \(q=11\) n, x \(8+ax\) 3 permutes \(F_ q\) if and only if \(n=1\) and \(a=2,4,7,9\); and if \(q=13\) n, then for any \(a\in F_ q\), x \(8+ax\) 5 is not a PP of \(F_ q\).
Reviewer: G.L.Mullen

MSC:
11T06 Polynomials over finite fields
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References:
[1] Lidl, Finite Fields 20 (1983)
[2] Mollin, Internat. J. Math. Sci. 10 pp 536– (1987)
[3] DOI: 10.2307/2004010 · Zbl 0141.03303 · doi:10.2307/2004010
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