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We discuss complete mapping polynomials of finite fields, which are a special class of permutation polynomials. Complete mapping polynomials of small degree are classified. Results are obtained on a class of complete mapping binomials and on permutation binomials. 1980 Mathematics subject classification (Amer. Math. Soc): 12 C 05.doi:10.1017/s1446788700018346 fatcat:wkcko5v4cfg2bhzy2u7yote5di