Why the q th power of a root is again a root？

Question

If $f(x)$ is a q-polynomial over $F_q$, q is prime, the roots of $f(x)$ form a vector space over $F_q$. Why the qth power of a root is again a root?

Answer 1

If $\alpha$ is a root of $f$, then $0=f(\alpha)^q=f(\alpha^q)$ because the map $x \mapsto x^q$ is an endomorphism of $\mathbb F_q$.

This map is called the Frobenius endomorphism.