TITLE: proving that there are infinitely many zero polynomials over a finite field
QUESTION [5 upvotes]: 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?

REPLY [9 votes]: There are finitely many functions over a finite field. There are infinitely many polynomials over a finite field. By the pigeonhole principle, it follows that there are infinitely many polynomials which represent the same function.
The difference of any two such polynomials will represent the zero function.