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Dummit and Foote's exercise 14.3.11 asks to prove that $f(x) = x^{p^{n}}-x+1$ is irreducible over $\mathbb{F}_{p}$ iff $n=1$ or $n=p=2$. To prove the 'only if' part, the exercise suggest to prove that if $\alpha$ is a root of $f(x)$, then so is $\alpha + a$ for every $a \in \mathbb{F}_{p^{n}}$. This implies that $\mathbb{F}_{p}(\alpha)$ contains $\mathbb{F}_{p^{n}}$ and that $[\mathbb{F}_{p}(\alpha):\mathbb{F}_{p^{n}}] = p$.

I've proven everything except that $[\mathbb{F}_{p}(\alpha):\mathbb{F}_{p^{n}}] = p$. Can anyone give me some ideas for this? Thanks.

Formal
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    It's a bad idea to denote both the polynomial and the characteristic by $p$. – azimut Sep 29 '13 at 21:30
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