How can I show for $M\in\text{GL}_d(\mathbb{Z}/p\mathbb{Z})$ with $\text{ord}(M)=p^n$ ($n$ a positive integer), that $1$ is an eigenvalue of $M$?

I would be grateful for any hint or advice.

Thank you!

  • 1
    $\begingroup$ Show that if $\alpha$ is an eigenvalue, then $\alpha^{p^n}=1$. $\endgroup$ – Lord Shark the Unknown Apr 22 at 8:32
  • $\begingroup$ Oh sorry! The $d$ isn't more specified on my problem sheet $\endgroup$ – TwoStones Apr 22 at 8:32
  • $\begingroup$ @LordSharktheUnknown Would you embed $M$ in $GL_d(F)$ the splitting field of its characteristic polynomial to show it has some eigenvector ? $\endgroup$ – reuns Apr 22 at 8:37
  • $\begingroup$ @LordSharktheUnknown Ok, I have shown that. But does from that already follow $\alpha = 1$? Sorry, if that's a trivial question, but I'm not familiar with modular arithmetic $\endgroup$ – TwoStones Apr 22 at 8:41
  • $\begingroup$ In characteristic $ p $, there is a marvelous formula called the freshman's dream : $ (x+y)^p = x^p + y^p $. Can you see how that can help ? $\endgroup$ – Max Apr 22 at 8:44

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