1
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ A big hint: Fermat's little theorem shows that it can't have any linear factors - why? $\endgroup$ – Steven Stadnicki Jul 26 '15 at 17:10
  • $\begingroup$ Several short ways here. My answer is possibly the most elementary, but others' arguments are much shorter! $\endgroup$ – Jyrki Lahtonen Jul 26 '15 at 17:12
  • 1
    $\begingroup$ Here's another thread with a short solution $\endgroup$ – Zev Chonoles Jul 26 '15 at 17:12
  • $\begingroup$ If $q$ is a prime power and $a\in\mathbb{F}_q$ is nonzero, then $x^q-x+a$ is also irreducible in $\mathbb{F}_q[x]$. $\endgroup$ – Batominovski Jul 26 '15 at 17:15
  • 1
    $\begingroup$ @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 $F$. $\endgroup$ – Batominovski Jul 26 '15 at 17:31