Question 74
Let $p$ be an odd prime number. We say a polynomial
$$
f(x) = \sum_{j=0}^{n} a_{j}x^{j} \text{ is } i\text{-residue if } \sum_{p-1 \lt j, j \gt 0} a_{j} \equiv i \pmod{p}.
$$
Show that $\{f(1), \dots, f(p-1)\}$ is a complete residue system modulo $p$ if and only if polynomials $f(x), \dots, f(x)^{p-2}$ are 0-residue and $(f(x))^{p-1}$ is 1-residue.
$$
f(x) = \sum_{j=0}^{n} a_{j}x^{j} \text{ is } i\text{-residue if } \sum_{p-1 \lt j, j \gt 0} a_{j} \equiv i \pmod{p}.
$$
Show that $\{f(1), \dots, f(p-1)\}$ is a complete residue system modulo $p$ if and only if polynomials $f(x), \dots, f(x)^{p-2}$ are 0-residue and $(f(x))^{p-1}$ is 1-residue.