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Problem 39 (Some Polynomial Factor Rings)
Complete the following:
- Use the first isomorphism theorem to prove that $\mathbb{Z}[x]/\left<x^2+1\right>$ is isomorphic to $\mathbb{Z}[i]$. From this we know that $\mathbb{Z}[x]/\left<x^2+1\right>$ is an integral domain and not a field.
- Is $\mathbb{Q}[x]/\left<x^2+1\right>$ an integral domain? a field?
- Is $\mathbb{R}[x]/\left<x^2+1\right>$ an integral domain? a field?
- Is $\mathbb{C}[x]/\left<x^2+1\right>$ an integral domain? a field?
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