Bijection polynomial maps on finite field

Question

It's well known that every function on a finite field is a polynomial funcion.

Let $\mathbb{F}$ be a field. It's easy to see a linear map $f:\mathbb{F} \mapsto \mathbb{F}$, $f(x) = ax+b$ is a bijective map iff $a \neq 0$.

Let $\mathbb{F}_q$ be a field. Is there any condition on polynomial of degree greater than 1 to be bijective map on $\mathbb{F}_q$? Are irreducible polynomials over $\mathbb{F}_q$ bijective map on $\mathbb{F}_q$?

Answer 1

If $f$ is a polynomial of degree $n\ge2$ which is bijective on $\Bbb F_q$, then $f$ has a zero, and so is reducible.