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$f$ is irreducible, How do I compute this $$\prod_{d\mid n}\prod_{f \in \mathbb{Z_p[x]},\, \deg(f) = d}f$$

I tried small examples for this:

So $n = 1, p = 2$. We have that this product is: $x(x + 1) = x^2 + x = x^2 - x$

$n = 2, p = 2$ We have that this product is : $x(x + 1)(x^2 + x + 1) = (x^2 + x)(x^2 + x + 1) = x^4 + x^2 + x + x^2 + 2 = x^4 + x = x^4 - x$

I'm looking at this and I'm guessing that this product is: $x^{p^n} - x$ but I have no idea how to show this in a general case. Would anyone have any suggestions how to do this?

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    $\begingroup$ Are you assuming that $f$ is monic? Yes -> duplicate. No -> ... reduce it to the monic case. $\endgroup$ – Martin Brandenburg Nov 4 '14 at 20:21
  • $\begingroup$ Woops, I didn't see that before I posted. Sorry everyone and thank you! $\endgroup$ – James Nov 4 '14 at 21:01
  • $\begingroup$ James, we close duplicates here. But if there is a step in the argument that you want to ask more about, you are more than welcome to do so. Either by editing this question (which then should be reopened in due course), or asking another one. In either case you should provide a link to that related older question. If you don't know how to do that, just ask. It probably is explained in the FAQ also. $\endgroup$ – Jyrki Lahtonen Nov 4 '14 at 21:21