# Roots of irreducible polynomial over finite field in splitting field

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.

• This is a recurring question. It is met so frequently that the first couple of years of the site's history nobody bothered to ask/answer it, because it is in most the books. The earliest incarnation I could find is this. As I happened to answer that I am a bit slow in calling this a duplicate even though I honestly think it is. Nov 10, 2021 at 9:11

$$f(x)=\sum^{n}_{k=0}a_kx^k$$. 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.