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Show that $x^p - x -1$ is irreducible over $\mathbb{F}_{p}$.

I've seen this polynomial (or some variation x^p -x -a) on several of our qualifying exams and in every case they ask you to show it is irreducible. I know you can show it is irreducible by showing that if $\alpha$ is a root of it then $\alpha+1$ is also a root and then showing that it is the minimal polynomial for its roots. This takes about half a page of writing to do and I am looking for a shorter way to prove this.

Anyone have a shorter way?

Tuo
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  • A big hint: Fermat's little theorem shows that it can't have any linear factors - why? – Steven Stadnicki Jul 26 '15 at 17:10
  • Several short ways here. My answer is possibly the most elementary, but others' arguments are much shorter! – Jyrki Lahtonen Jul 26 '15 at 17:12
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    Here's another thread with a short solution – Zev Chonoles Jul 26 '15 at 17:12
  • If $q$ is a prime power and $a\in\mathbb{F}_q$ is nonzero, then $x^q-x+a$ is also irreducible in $\mathbb{F}_q[x]$. – Batominovski Jul 26 '15 at 17:15
  • Weird. Marked as duplicate but when I was typing up the question that never came up as a "similar problem that might have my answer" – Tuo Jul 26 '15 at 17:17
  • @StevenStadnicki just because it doesn't have linear factors doesn't mean it's irreducible though – Tuo Jul 26 '15 at 17:18
  • @Batominovski: I don't think that's right. Check your source (and/or recalibrate your intuition :-). And look at this thread. – Jyrki Lahtonen Jul 26 '15 at 17:19
  • @Andrew: It often happens that you see better matches in the "Related" list (see the right margin). I don't know how the system works. The local search engine has problems with TeX-parts (may be it cannot use it at all?). Don't worry about it. – Jyrki Lahtonen Jul 26 '15 at 17:22
  • And, if you don't find those answers satisfactory @-ping me and explain what else you think could be simplified. Any moderator can reopen this with a single click, but they may not want to overrule me :-/ – Jyrki Lahtonen Jul 26 '15 at 17:24
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    @Jyrki Lahtonen, you are right. I misremembered the problem. I know there is a similar problem that doesn't require the field to be $\mathbb{F}_p$. Let me state it here. Let $F$ be a field of characteristic $p$ with $p$ being prime. Then, $x^p-x+c$ is irreducible in $F[x]$ if and only if $x^p-x+c$ has no root in $F$. – Batominovski Jul 26 '15 at 17:31

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