Are they regarded as the same polynomial in $F_p$?

Question

Let $p$ be prime. Let $p_1(x)=x^p-x$, $p_2(x)=0$ be two polynomials in $F_p[x]$. We know that $p_1=p_2$ as functions, by Fermat's little theorem. However, can I say that $p_1(x)$ is the zero polynomial?

Answer 1

No. The elements of a polynomial ring like $\mathbb{F}_p[x]$ are polynomials as formal expressions, not as functions. In other words, two polynomials are defined to be the same only when all of their coefficients are the same. (Or if you prefer, a polynomial is defined to be its coefficients: an element of $\mathbb{F}_p[x]$ is technically a function $f:\mathbb{N}\to \mathbb{F}_p$ such that $f(n)=0$ for all but finitely many $n\in\mathbb{N}$. We normally write such a function as the "polynomial expression" $f(0)+f(1)x+f(2)x^2+f(3)x^3+\dots$.)

Answer 2

No. They give the same functions $\Bbb Z_p\to\Bbb Z_p$ (the zero function) but they are not the same polynomials.