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Is there any way to motivate why certain factor rings of $\mathbb{Z}, \mathbb{Z}[i]$, etc., to a prime power have non-cyclic unit groups? For example, the only such non-cyclic unit groups of factor rings of $\mathbb{Z}$ are prime powers of $2$. I think this might be because $\mathbb{Z}$ has its $2$-nd roots of unity but the situation in general rings seems much more complicated.

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There a lot more quotients of $\mathbb{Z}$ that have non-cyclic unit groups besides the ones moding out by $2^n$ with $n\gt 2$; for example, if $p$ and $q$ are distinct odd primes, then the unit group of $\mathbb{Z}/pq\mathbb{Z}$ is not cyclic, since by the Chinese Remainder Theorem it follows that it will be isomorphic to $(\mathbb{Z}/(p-1)\mathbb{Z})\times(\mathbb{Z}/(q-1)\mathbb{Z})$, which is not cyclic. – Arturo Magidin Jul 2 '12 at 20:37
@ArturoMagidin That's true, my question is for prime powers (or prime ideal powers). I just wonder if there's a way to predict that $2$ behaves strangely in $\mathbb{Z}$ in this regard: is it related to the unit group of $\mathbb{Z}$? is it because $2$ ramifies in some field that other primes don't? Or is it just there? – Anonymouse Jul 2 '12 at 21:13
As another example, the primes in $\mathbb{Z}[i]$ whose powers give cyclic unit groups are exactly the factors of primes in $\mathbb{Z}$ which are $\equiv 1 \mod 4$. I've seen the proof, but I don't see any intuition for it. – Anonymouse Jul 2 '12 at 21:16

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