# proving that there are infinitely many zero polynomials over a finite field

Given a finite field F with c elements, how would one prove that there are infinitely many polynomials that represent the zero function over the field F? I know we have infinitely many polynomials over this field, isn't it straightforward that by choosing all the coefficients to be zero all those infinitely many polynomials are zero functions?

• Do you understand the difference between polynomial and polynomial function? Commented Mar 15, 2017 at 0:10
• Those polynomials all coincide. I think you want examples like $x^p-x, x^{p^2}-x$ and so on.
– lulu
Commented Mar 15, 2017 at 0:10

Denote by $f_1, f_2, \dots, f_n$ the elements of the finite field $\mathbb{F}$. Then $$f(x) = (x-f_1)(x-f_2)\cdots (x-f_n) \not = 0 \in \mathbb{F}[x],$$ even though the function $\mathbb{F} \ni a \mapsto f(a)$ is identically zero.
The powers $f(x)^n$, $n\in \mathbb{N}$, provide you with infinitely many different non-zero polynomials representing the zero function.
• Nice answer! I suppose, if the finite field is $\mathbb F_{p^n}$, then your polynomial is $f(x) = x^{p^n} - x$. Commented Mar 15, 2017 at 0:50