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Is $U(2^{n})$ isomorphic to $\mathbb{Z_{2}} \bigoplus \mathbb{Z_{2^{n-2}}}$ if $n\geq3$?

And is $U(p^{n})$ isomorphic to $\mathbb{Z_{p^{n}-p^{n-1}}}$ where $p$ an odd prime?

I'm really wondering about its proof.

It's not my homework. I used this fact when doing my homework. I don't have to prove this fact for my homework. This question is just for curiosity.

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Related to and possible duplicate of – lhf Aug 16 '11 at 11:11
@Kim Hee yeon: I have merged your unregistered account with your registered account (this one). If you encounter any further trouble logging in, please let one of the moderators know (via a post on meta, or via a comment) – Zev Chonoles Aug 16 '11 at 23:44

Yes, this is the well-known structure theorem for units in modular rings. Almost all number theory books contain a proof. See, for instance, Chapter 4 of LeVeque's Fundamentals of Number Theory.

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