(From Milne) Splitting field over a finite field.

Question

I have stuck on this question for some time but when I check the solution bank I find the proof is omited. May I please ask for some explaination? Any help is appreciated!

Answer 1

Here are some ideas for you to mull...and understand and prove:

=== Prove that the set of all roots of $\;x^{p^r}-x\in\Bbb F_p[x]\;$ (these roots "live" in some algebraic closure of $\;\Bbb F_p\;$ , or if you prefer: in some extension field of $\;\Bbb F_p\;$) is a field $\;\Bbb F_{p^r}\;$ with $\;p^r\;$ elements under usual addition and multiplication modulo $\;p\;$ .

=== Show $\;\dim\left(\Bbb F_{p^r}\right)_{\Bbb F_p}=r\;$ and deduce that $\;\Bbb F_{p^r}=\Bbb F_p(\alpha)\;$ , for some element $\;\alpha\in\Bbb F_{p^r}\;$ whose minimal polynomial in $\;\Bbb F_p[x]\;$ has degree $\;r\;$

=== In general, suppose that for some $\;g(x)\in\Bbb Z[x]\;$ we have that $\;h(x):=g(x)\mod p\in\Bbb F_p[x]\;$ is irreducible. Then $\;g(x)\;$ is irreducible in $\;\Bbb Z[x]\;$ and thus also (Gauss...?) in $\;\Bbb Q[x] \;$