IMC 2009 · Problem 4

Let $p$ be a prime number and $\mathbb{F}_p$ be the field of residues modulo $p$. Let $W$ be the smallest set of polynomials with coefficients in $\mathbb{F}_p$ such that

• the polynomials $x + 1$ and $x^{p-2} + x^{p-3} + \dots + x^2 + 2x + 1$ are in $W$, and

• for any polynomials $h_1(x)$ and $h_2(x)$ in $W$ the polynomial $r(x)$, which is the remainder of $h_1(h_2(x))$ modulo $x^p - x$, is also in $W$.

How many polynomials are there in $W$?