polynomials in $\mathbb F_p$

Question

Let $p$ be a prime number and denote by $\mathbb F_p$ the field of integers modulo $p$. Like before, we can consider polynomials with coefficients in $\mathbb F_p$. Then, the polynomial $X^p − X$ is non-zero and its polynomial function $x^p − x$ is zero, by little Fermat’s theorem.

Can someone please explain what the above statement means? Why is the polynomial $X^p - X$ non zero, but the polynomial function $x^p -x$ is zero? what is it referring to? If there are any references online, I'd appreciate it. I searched online about the above statement but couldn't find anything relevant. Thanks!

Answer 1

A polynomial exist by itself without the need of evaluate it at some point, if you want you can interpret it as a formal expression. When consider the possibility of evaluate , the polynomial become a funcion. In this case the zero map and the polynomial X^p-X define the same map, in the field with p elements.

Answer 2

Here is q definition of polynomials. For a ring $R$, we define the polynomial ring $R[X]$ to be the set of all sequences $(a_0, a_1, a_2,\ldots)$ for $a_i$ in $R$, where finitely many $a_i$ are nonzero. Furthermore, we define a ring structure on $R[X]$ by $$ (a_0,a_1,\ldots) + (b_0,b_1,\ldots) = (a_0+b_0,a_1+b_1,\ldots) $$ $$ (a_0,a_1,a_2,\ldots)\cdot (b_0,b_1,b_2\ldots) = (a_0b_0, a_0b_1 + a_1b_0, a_2b_0 + a_1b_1 + a_2b_0,\ldots) $$ this notation is pretty unwieldy, so we usually write the element $(a_0,a_1,a_2,\ldots)$ as $a_0 + a_1X + a_2X^2 + \ldots$.

If you want, you can ignore everything written above. The important point is that a polynomial $a_0 + a_1X + \ldots + a_nX^n$ is defined by the numbers $a_i$; a polynomial literally is its sequence of coefficients. Therefore, the zero polynomial corresponds to the sequence $(0, 0, 0,\ldots)$, and the polynomial $X^p - X$ corresponds to the sequence $(-1, 0, 0, \ldots, 0, 1, 0, \ldots)$ where the $1$ is in the $p^{\text{th}}$ position. These sequences are different, so the polynomials are different. In other words, the polynomials $0$ and $X^p - X$ have different coefficients, hence they are distinct polynomials.

On the other hand, we might want to define the function $f:\mathbb{F}_p\to\mathbb{F}_p$ by $f(x) = x^p - x$, which we generally refer to as the "evaluation" of the polynomial at $x \in \mathbb{F}_p$. It turns out that this evaluation is actually zero at every point in $\mathbb{F}_p$, and hence every element of $\mathbb{F}_p$ is a root of the polynomial $X^p - X$, but the actual polynomial is not zero.