(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that [itex]x^{p^n}-x[/itex] is the product of all the monic irreducible polynomials in [itex]\mathbb{Z}_p[x][/itex] of a degree d dividing n.

2. Relevant equations

3. The attempt at a solution

So, I want to prove that the zeros of all such monic polynomials are also zeros of [itex]x^{p^n}-x[/itex] and vice versa. I cannot do either, unfortunately. We know that the elements of GF([itex]p^n[/itex]) are precisely the zeros of [itex]x^{p^n}-x[/itex]. And we know that if f(x) is a monic polynomial of degree m in [itex]\mathbb{Z}_p[x][/itex], then when you adjoin any of its zeros to [itex]\mathbb{Z}_p[/itex], you get a field with p^m elements whose elements are precisely the zeros of [itex]x^{p^m}-x[/itex]. So, I guess if [itex]\alpha[/itex] is a zero of [itex]x^{p^n}-x[/itex], then does the irreducible monic polynomial that [itex]\alpha[/itex] is a zero of need to be a factor of [itex]x^{p^n}-x[/itex]?

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# Product of monic polynomials

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