The Carlitz conjecture on even-degree exceptional polynomials

Let $q$ be a prime power. A polynomial $f(X)\in\mathbb{F}_q[X]$ is exceptional if it induces a bijection on $\mathbb{F}_{q^k}$ for infinitely many positive integers $k$. Carlitz conjecture. There exists no even-degree exceptional polynomial over $\mathbb{F}_q$ for any odd prime power $q$. The conjecture concerns restrictions on the degrees of exceptional polynomials over finite fields and was fully proved by Fried, Guralnick, and Saxl using the classification of finite simple groups.