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.