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Problem 35 (When Is A Polynomial Factor Ring An Integral Domain)

Let $R=\mathbb{Z}[x]$. Suppose that $A=\left<p(x)\right>$ is a principle ideal generated by a single polynomial.

  1. Find a polynomial $p(x)$ of degree 2 such that $R/A$ is not an integral domain. Then find one of degree 3 and degree 4.
  2. Find a polynomial $p(x)$ of degree 3 such that $R/A$ is an integral domain.
  3. If you know that $R/A$ is not an integral domain, what do you know about $p(x)$?
  4. State a simple condition on $p(x)$ that is equivalent to $R/A$ being an integral domain.

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