Question 87
We call a monic polynomial $P(x) \in \mathbb{Z}[x]$ square-free mod $n$ if there are no polynomials $Q(x), R(x) \in \mathbb{Z}[x]$ such that $Q$ being non-constant and
$$
P(x) \equiv Q(x)^2 R(x) \pmod{n}.
$$
Given a prime $p$ and integer $m \ge 2$. Find the number of monic square-free mod $p$ polynomials $P(x)$ with degree $m$ and coefficients in $\{0, 1, 2, 3, \dots, p-1\}$.
$$
P(x) \equiv Q(x)^2 R(x) \pmod{n}.
$$
Given a prime $p$ and integer $m \ge 2$. Find the number of monic square-free mod $p$ polynomials $P(x)$ with degree $m$ and coefficients in $\{0, 1, 2, 3, \dots, p-1\}$.