I'm my study of Galois theory I have been struggling with the following proposition without much success:

The polynomial $X^{p^n}-X$ is precisely the product of all the distinct irreducible polynomials in $\mathbb{F}_p[X]$ of degree $d$ where $d$ runs through all divisors of $n$

For example, $X^{p^6}-X$ is factorized by a polynomial of degree 1, 2, 3 and 6? It is posible to obtain them explicitly?

  • $\begingroup$ You probably mean mod $p$. And the answer depends in $p$. Start here: Cyclotomic polynomial - Wikipedia $\endgroup$ – lhf Dec 6 '18 at 16:59
  • $\begingroup$ @lhf yep edited for $\mathbb{F}_p[X]$. I'm going to read about your link. Thanks. $\endgroup$ – UnPerrito Dec 6 '18 at 17:02
  • $\begingroup$ If $\mathbb{F}_p(\alpha) = \mathbb{F}_{p^6}$ then what is the minimal polynomial of $\alpha$ over $\mathbb{F}_p$ ? $\endgroup$ – reuns Dec 6 '18 at 17:12
  • $\begingroup$ @reuns as far I know, it must be a polynomial of degree 6 $\endgroup$ – UnPerrito Dec 6 '18 at 17:18
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    $\begingroup$ Finding those factors is possible but somewhat taxing. If the only input you are given is: Factor $x^{p^n}-x$! then I would go with the Cantor-Zassenhaus algorithm. Works for any polynomial over any finite field! But, if $p$ and/or $n$ are largish it will be a lot of work. This toy example I did when learning about C-Z will show what to expect. $\endgroup$ – Jyrki Lahtonen Dec 6 '18 at 18:58

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