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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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    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.2012-07-02

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