# factors of $X^{p^n}-X$

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?

• You probably mean mod $p$. And the answer depends in $p$. Start here: Cyclotomic polynomial - Wikipedia – lhf Dec 6 '18 at 16:59
• @lhf yep edited for $\mathbb{F}_p[X]$. I'm going to read about your link. Thanks. – UnPerrito Dec 6 '18 at 17:02
• If $\mathbb{F}_p(\alpha) = \mathbb{F}_{p^6}$ then what is the minimal polynomial of $\alpha$ over $\mathbb{F}_p$ ? – reuns Dec 6 '18 at 17:12
• @reuns as far I know, it must be a polynomial of degree 6 – UnPerrito Dec 6 '18 at 17:18
• 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. – Jyrki Lahtonen Dec 6 '18 at 18:58