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?

Consider counting the total number of monic polynomials with roots, for fixed degree : say for degree $3$ , you can have $(x-r_1)(x-r_2)(x-r_3)$ (notice that you need to
consider that different permutations of $r_1,r_2,r_3$ will generate the same polynomial). And the overall number of monic polynomials will be $x^3+ax^2+bx+c$ , where (hint) $(a,b,c)$ is in $\mathbb F^3$.
EDIT: The identity I had in mind was: for degree q<|$\mathbb F$| , there are $\mathbb FCq$ ways of having all different roots, then, for having 1 repeat....1 way of having
all roots equal, compared with |$\mathbb F^3$|. Even for $q=2$, we have $\mathbb FC2$+|$\mathbb F$| , versus |$\mathbb F^2$|, where the $C$ means "choose".
