Find a non-trivial polynomial which has all integers as zeros modulo $p$

Question

This is an exercise from the book "Einführung in die algebraische Zahlentheorie" by Alexander Schmidt. Let $p$ be a prime number. And let $f \in \mathbb{Z}[X]$ be a polynomial with integer coefficients such that it is not the zero polynomial modulo $p$ ($f \not\equiv 0 \pmod p$). But such that $f(a)\equiv 0 \pmod p$ for all $a\in \mathbb{Z}$.

Answer 1

Let's assume that $f\in F_p[X]$ is a non-zero polynomial with all $a\in F_p$ as roots. When $a\in F_p$ is a root, $(x-a)$ is a factor, thus $f(x)=g(x)\prod_{i=0}^{p-1} (x-i)$ for some non-zero $g\in F_p[X]$. We also see that every polynomial on the form $g(x)\prod_{i=0}^{p-1} (x-i)$ has every $a\in F_p$ as a root. We conclude that the polynomials are precisely the ones on the form: $$g(x)\prod_{i=0}^{p-1} (x-i),$$ where $g\not\equiv 0$.