TITLE: When does the underlying map of a polynomial induce a permutation on $\mathbb{Z}/p\mathbb{Z}$?
QUESTION [8 upvotes]: For example, the underlying function of the polynomial
$$f(x)=4x^2-3x^7$$ induces a permutation on $\mathbb{Z}/11\mathbb{Z}$, though I only know the proof by "brutal force" (is there a cleverer proof?).
In general, is there a criterion determining if a polynomial $f(x)$ induces a permutation on $\mathbb{Z}/p\mathbb{Z}$?
Edit: I am not even sure if it matters whether $p$ is prime or not but I will tag this question "finite fields" for now. Does it matter?

REPLY [4 votes]: Wikipedia has the answer:

Let $\mathbb F_q$ be a finite field of characteristic $p$. Then $f \in \mathbb F_q[x]$ is a permutation polynomial iff $f$ has exactly one root in $\mathbb F_q$ and for each integer $t$ with $1 \le t \le q-2$ and $t\not\equiv 0 \bmod p$, the degree of $f(x)^t \bmod (x^q-x)$ is at most $q-2$.

This is known as Hermite's criterion.
See also these surveys:

Permutation polynomials over finite fields — A survey of recent advances (2015)
Permutation polynomials of finite fields (2012)