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How can we determine whether the quotient rings $R/I$ where where $R=\mathbb Z[x], \mathbb Z_2[x], \mathbb Z_3[x]$ and $I=(x^2+1), (x^2+x+1)$ are fields, principal ideal domains, unique factorization domains or integral domain? (Of course they are all integral domains, but I am looking for the strictest condition.) Thank you.

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    Are you quite sure that they’re all integral domains? What do you get when you simplify $(x+1)^2$ in $\Bbb{Z}_2[x]$? What about $(x+2)^2$ in $\Bbb{Z}_3[x]$?2012-03-04

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