- #1

ehrenfest

- 2,020

- 1

## Homework Statement

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.

## Homework Equations

## 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]?

Last edited: