Finding the monic generator of ideals of polynomials

Question

Given $F$ a field, and $F^F$, the set of functions from $F$ to $F$, and define an evaluation map $e : F[x]\to F^F$ which sends a polynomial to the function which is computed using the polynomial as formula for computation. Given that the kernel of $e$ is an ideal of $F[x]$, how would I show that the monic generator of this ideal is $x^q - x$, where $F$ is a finite field with $q$ elements?

Answer 1

Write $f(x)=(x^q-x)g(x)+r(x)$ with $\deg r<q$. If $f(a)=0$ for all $a\in F$, then $r(a)=$ for all $a\in F$. How many distinct roots can have a polynomial (of degree at least one) with coefficients in a field?