Question 16
Let $p \gt 2$ be a prime number. $\mathbb{F}_p[x]$ is defined as the set of all polynomials in $x$ with coefficients in $\mathbb{F}_p$ (the integers modulo $p$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^k$ are equal in $\mathbb{F}_p$ for each nonnegative integer $k$. For example, $(x+2)(2x+3) = 2x^2 + 2x + 1$ in $\mathbb{F}_5[x]$ because the corresponding coefficients are equal modulo $5$.
Let $f, g \in \mathbb{F}_p[x]$. The pair $(f, g)$ is called compositional if
$$
f(g(x)) \equiv x^{p^2} - x
$$
in $\mathbb{F}_p[x]$. Find, with proof, the number of compositional pairs (in terms of $p$).
Let $f, g \in \mathbb{F}_p[x]$. The pair $(f, g)$ is called compositional if
$$
f(g(x)) \equiv x^{p^2} - x
$$
in $\mathbb{F}_p[x]$. Find, with proof, the number of compositional pairs (in terms of $p$).