$F$ be a finite field , then are there infinitely many polynomials $f(x) \in F[x]$ such that $f(a)=0 , \forall a \in F$ ?

Question

Let $F$ be a finite field , then is it true that there are infinitely many polynomials $f(x) \in F[x]$ such that $f(a)=0 , \forall a \in F$ ?

Answer 1

If the field has $q$ elements, then $x^q-x$ works. For if $a$ is non-zero, then $a^{q-1}-1=0$. Multiplying by $x$ ensures that the polynomial is also $0$ when $a=0$.

It follows that $x^{q+k}-x^{k+1}$ also works for any positive integer $k$.