Geodesic
On Exceptional Polynomials
Canadian mathematical bulletin, Tome 11 (1968) no. 2, pp. 279-282

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Let f(x) be a polynomial of degree d ≥ 2 defined over the finite field kq with q = pn elements. Let If f*(x, y) has no irreducible factor over kq which is absolutely irreducible, f is called an exceptional polynomial [1]. Davenport and Lewis have noted that when d is small compared with p, a permutation (substitution) polynomial is necessarily an exceptional polynomial. It is the purpose of this paper to prove the converse; that is, we will show the existence of a constant a(d), depending only on d, such that if f(x.) is an exceptional polynomial over kq, where p ≥ a(d), then f(x) is a permutation polynomial. 
Williams, Kenneth S. On Exceptional Polynomials. Canadian mathematical bulletin, Tome 11 (1968) no. 2, pp. 279-282. doi: 10.4153/CMB-1968-033-1
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     title = {On {Exceptional} {Polynomials}},
     journal = {Canadian mathematical bulletin},
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     year = {1968},
     volume = {11},
     number = {2},
     doi = {10.4153/CMB-1968-033-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1968-033-1/}
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