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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
##### Keywords:
finite field; permutation polynomial
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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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