In mathematics, the CarlitzâÂÂWan conjecture classifies the possible degrees of exceptional polynomials over a finite field F<sub>q</sub> of q elements. A polynomial f(x) in F<sub>q</sub>[x] of degree d is called exceptional over F<sub>q</sub> if every irreducible factor (differing from x â y) or (f(x) â f(y))/(x â y)) over F<sub>q</sub> becomes reducible over the algebraic closure of F<sub>q</sub>. If q > d<sup>4</sup>, then f(x) is exceptional if and only if f(x) is a permutation polynomial over F<sub>q</sub>.
The CarlitzâÂÂWan conjecture states that there are no exceptional polynomials of degree d over F<sub>q</sub> if gcd(d, q â 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). The general form of the CarlitzâÂÂWan conjecture was proposed by Daqing Wan (1993) and later proved by Hendrik Lenstra (1995).