Properties of polynomial in finite field

Question

Let $f \in F_p[x]$.

Prove that polynomials $f(x), f(x + 1), \dotso, f(x + p - 1)$ are either pairwise distinct or they all coincide.

I think the right approach may involve Lagrange polynomials.

Answer 1

Let $0<k<p$. Then $$f(x)=f(x+k)$$ implies that $$f(x)=f(x+ak)$$ for all $a\in \mathbb N$ (because $f(x+(a-1)k)=f((x+(a-1)k)+k)$, so we can use an inductive proof). For any $0\leq m<p$, there is an $a$ such that $$ak\equiv m\pmod p$$ Thus $f(x)= f(x+m)$ for all $m$, and we are done.