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A simple proof of a conjecture about complete mapping over finite fields. (Chinese. English summary) Zbl 1007.11075
Summary: In [J. Aust. Math. Soc., Ser. A 33, 197-212 (1982; Zbl 0495.12018)] H. Niederreiter and K. H. Robinson conjectured that if $$F=F_q$$ is a finite field of $$q$$ elements with $$\operatorname {ch}F=2$$, $$q>3$$, and $$f(x)$$ is a complete mapping polynomial of $$F$$ (i.e. both $$f(x)$$ and $$f(x)+x$$ are permutation polynomials of $$F$$), then the reduction of $$f(x) \operatorname {mod} (x^q-x)$$ has degree $$\leq q-3$$. In [J. Aust. Math. Soc., Ser. A 41, 336-338 (1986; Zbl 0607.12009)], D. Wan proved this conjecture to be true. In this note, by an application of the 2-adic number field, the authors give a simple proof for this conjecture.

##### MSC:
 11T06 Polynomials over finite fields 12E20 Finite fields (field-theoretic aspects)