Английская Википедия:Carlitz–Wan conjecture
In mathematics, the Carlitz–Wan conjecture classifies the possible degrees of exceptional polynomials over a finite field Fq of q elements. A polynomial f(x) in Fq[x] of degree d is called exceptional over Fq if every irreducible factor (differing from x − y) or (f(x) − f(y))/(x − y)) over Fq becomes reducible over the algebraic closure of Fq. If q > d4, then f(x) is exceptional if and only if f(x) is a permutation polynomial over Fq.
The Carlitz–Wan conjecture states that there are no exceptional polynomials of degree d over Fq if gcd(d, q − 1) > 1.
In the special case that q is odd and d is even, this conjecture was proposed by Leonard Carlitz (1966) and proved by Fried, Guralnick, and Saxl (1993).[1] The general form of the Carlitz–Wan conjecture was proposed by Daqing Wan (1993)[2] and later proved by Hendrik Lenstra (1995)[3]
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