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Exercise 13.5.6 (Dummit & Foote): Prove that $x^{p^n-1}-1 =\prod\limits_{\alpha\in\mathbb{F}^\times_{p^n}} (x-\alpha)$.

How would I go about this? From what I have seen online they begin with $x^{p^n}-x$, I'm not really sure why they start with that. Any help with the whole problem will be appreciated.

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  • Start from the question of which $\alpha$ satisfy $\alpha\in\mathbb{F}^\times_{p^n}$. – J.G. Apr 07 '21 at 15:16
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    There are some helpful posts about this topic, e,g, here. The answer also depends on what you already know and which can be assumed without proof. So you should give more context. – Dietrich Burde Apr 07 '21 at 15:23

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Hint: $\mathbb{F}^\times_{p^n}$ is a multiplicative group of order $p^n-1$. By Lagrange's theorem, $\alpha^{p^n-1}=1$ for all $\alpha \in\mathbb{F}^\times_{p^n}$.

lhf
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