# Eigenvalue of matrix over (Z/pZ)

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!

• Show that if $\alpha$ is an eigenvalue, then $\alpha^{p^n}=1$. – Lord Shark the Unknown Apr 22 at 8:32
• Oh sorry! The $d$ isn't more specified on my problem sheet – TwoStones Apr 22 at 8:32
• @LordSharktheUnknown Would you embed $M$ in $GL_d(F)$ the splitting field of its characteristic polynomial to show it has some eigenvector ? – reuns Apr 22 at 8:37
• @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 – TwoStones Apr 22 at 8:41
• 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 ? – Max Apr 22 at 8:44