# $f(x)\in \mathbb F_p[X]$ irreducible. Then a splitting field for $f(x)$ has $p^n$ elements

Let $$\mathbb F_p$$ be a field with $$p$$ elements and consider f$$f(x)\in \mathbb F_p[X]$$ irreducible of degree $$n$$. Then a splitting field for $$f(x)$$ has $$p^n$$ elements.

Write $$S_f$$ for the splitting field of $$f(x)$$. Basically I have to show that $$[S_f:\mathbb F_p]=n$$ and the conclusion will follow. But I can't find a way to show that. I already deduced that $$f(x)$$ is separable and thus has $$n$$ different roots in $$S_f$$, but it does not seem helpful.

Any ideas? Thanks!

Write $$\;K:=\Bbb F_p[X]/\langle f(X)\rangle\;$$ . This is a field which contains (an isomorphic copy of) $$\;\Bbb F_p\;$$ and of dimension $$\;n=\deg f\;$$ over it . Since any finite extension of a finite field is Galois, we get that $$\;K\;$$ is normal over $$\;\Bbb F_p\;$$ and is thus a splitting field of any irreducible polynomial in $$\;\Bbb F_p[X]\;$$ having a root in it, so we deduce $$\;K=S_f\;$$ .