# Unique factorization domain

Are $\mathbb{Z}[x]/(x^N-1)$, $\mathbb{Z}_a[x]/(x^N-1)$, $\mathbb{Z}_p[x]/(x^N-1)$ UFDs, where $a$ is composite and $p$ is prime? $N$ may be prime or composite.

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First you might want to think about which of them are D's... –  Cam McLeman Mar 7 '12 at 16:12

Hint $\rm\ \mathbb Z_m[x]/(x^n-1)\:$ domain $\rm\:\Rightarrow\: m\:$ prime (or $0$) in $\rm\mathbb Z,\:\ x^n-1\:$ prime in $\rm\:\mathbb Z_m[x]\:$ $\Rightarrow$ $\rm\: n=1\ (or\:\ 0).\:$

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