0
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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. $\endgroup$ Nov 10, 2021 at 9:11

1 Answer 1

1
$\begingroup$

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

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .