Roots of irreducible polynomial over finite field in splitting field

Question

Let $\mathbb F_q$ be the finite field with $q$ elements. And $f\in \mathbb F_q[x]$ is an irreducible polynomial. I am trying to prove $L=\mathbb F_q[x]/(f)$ is the splitting field of $f$. And one of the problem says, if $\alpha\in L$ is a root of $f$, then $\alpha^{q^i}$ are also root of $f$ for $i=1,2,\cdots,n$. This looks quite easy but I am somehow stuck. I was thinking some analogue of Fermat's little theorem for polynomial but did really work it out. Any hint is appreciated.

Answer 1

$f(x)=\sum^{n}_{k=0}a_kx^k$. Then we have $f(x)^q=\sum^{n}_{k=0}(a_k)^q(x^q)^k$ but since $a_k\in \mathbb{F}_q$, we have $a_k^q=a_k$. Then repeat the argument repeatedly.