polynomial over a finite field Show that in a finite field $F$ there exists $p(x)\in F[X]$ s.t $p(f)\neq 0\;\;\forall f\in F$
Any ideas how to prove it?

Take some element $\alpha_1\in F$
Then consider $f_1(x)=(x-\alpha_1)+1$.. What would be $f_1(\alpha_1)$?
Soon you will see that $f(\alpha_1)$ is non zero but may probably for some $\alpha_2$ we have $f_1(\alpha_2)=0$
Because of this i would now try to include $(x-\alpha_2)$ in $f_1(x)$ to make it 
$f_2(x)=(x-\alpha_1)(x-\alpha_2)+1$.. What would be $f_1(\alpha_1),f
_2(\alpha_2)$?
keep doing this until you believe that resultant function does not have a root in $F$.
You have two simple questions :


*

*will the resultant be a polynomial in general if you repeat this steps.

*How do you make sure that no element in field is a root of resultant

