Carlitz–Wan conjecture

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