SCHWEITZER 1951 Problem 10

Let $f(x)$ be a polynomial with integer coefficients and let $p$ be a prime. Denote by $z_1,...,z_{p-1}$ the $(p-1)$ th complex roots of unity. Prove that $f(z_1)\cdots f(z_{p-1})\equiv f(1)\cdots f(p-1) \pmod{p}$ .