Problems from finite fields theory

Question

The problem is to find two distinct polynomials over $F_p$ which coincide as functions $F_p \to F_p$. So far I don't know where to begin. And the second question how one can check whether given polynomial $f(x) \in Q[x]$ is irreducible? (maybe it's matter: all coefficients is in $Z$) I know only sufficient condition.

Answer 1

For any field $\;\Bbb F_p=\{0,1,2,...,p-1\}\pmod p\;$ , the non-trivial polynomial of degree $\;p\;$

$$f(x)=x(x-1)(x-2)\cdot\ldots\cdot(x-(p-1))$$

is the same, as a function $\;:\Bbb F_p\to\Bbb F_p\;$, as the zero polynomial $\;g(x)\in\Bbb F_p[x],\;\;g(x)\equiv 0\;$ , of degree $\;-1\;$ (or $\;-\infty\;$ , depending on the use).

The second question has no definite answer in the general case: for polynomials of degree $\;\le3\;$ though, a polynomial is irreducible over any field $\;\Bbb F\;$ iff it has no root there, and there are some other rather restricted criteria for very specific cases.