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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!

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Some pointers:

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 $\;K=S_f\;$ .

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