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Problem 33 (Some Polynomial Factor Rings Of $\mathbb{Z}[x]$)

Let $R=\mathbb{Z}[x]$. Consider the three ideals $A=\left<x^2\right>$, $B=\left<x^2- 1\right>$, $C=\left<x^2+1\right>$.

  1. Show that $R/A=\{a+bx+A\mid a,b\in\mathbb{Z}\}$.
  2. Show that $R/A$ is not an integral domain.
  3. Is $R/B$ an integral domain? Prove your answer.
  4. Is $R/C$ an integral domain? Prove your answer.

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