Let $p$ a prime number. Show that the polynomial $X^p-X-1 \in F_{p}[X]$ is irreducible using this hint:

If $a$ is a root of $X^p-X-1$, show that $a^{p^{p}}=a$, who is the extension $F_{p}(a)$?.

Can anyone help me to show this? How I can use this hint?

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    $\begingroup$ The irreducibility claim is not even true (check $p=5$). $\endgroup$ – darij grinberg Aug 28 '15 at 21:28
  • $\begingroup$ i am sorry it was $p$ and not $2$ $\endgroup$ – Legolas Aug 28 '15 at 21:58
  • $\begingroup$ Then this is the umpteenth reincarnation of the same question. See here for many different approaches. $\endgroup$ – Jyrki Lahtonen Aug 28 '15 at 22:07
  • $\begingroup$ I feel like adding something I read about in an old paper by Artin himself today. His argument was as follows. "Let $\gamma$ be one zero of your polynomial. The other zeros are then $\gamma+1,\gamma+2,\ldots$. Any one them generates the same extension field of $\Bbb{F}_p$. Therefore all the irreducible factors have the same degree..." So much prettier than my answer in the linked thread. Other answerer came closer :-) $\endgroup$ – Jyrki Lahtonen Aug 28 '15 at 22:10
  • $\begingroup$ @Jyrki Lahtonen I saw your answer for the post that you send me, I am wondering how i can show it with the specific hint $\endgroup$ – Legolas Aug 28 '15 at 22:15

For odd primes $p$, attempt to solve $X^2 - X - 1 = 0$ using the quadratic formula. Your polynomial is irreducible iff it has no root iff $5$ is not a square mod $p$.

I'll leave the $p=2$ case to you.


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