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Problem 26 (Factor Rings Of $\mathbb{Z}$ And $\mathbb{Z}[x]$)
Answer the following questions as they pertain to the integral domains $\mathbb{Z}$ and $\mathbb{Z}[x]$. You are welcome to rapidly make claims about factor rings, without proof, as we have seen many of these as factor groups in the past.
- We know that the ideals of $\mathbb{Z}$ are of the form $n\mathbb{Z} = \left< n\right>$, the set of multiples of $n$.
- For which $n$ is the factor ring $\mathbb{Z}/\left<n\right>$ an integral domain?
- For which $n$ is the factor ring $\mathbb{Z}/\left<n\right>$ a field?
- Now consider the ring of polynomials $\mathbb{Z}[x]$. The ideal $\left<n\right>$ is now the set of polynomials whose coefficients are multiples of $n$. It should not be a surprise that $\mathbb{Z}[x]/\left<n\right> \approx \mathbb{Z}_n[x]$.
- For which $n$ is the factor ring $\mathbb{Z}[x]/\left<n\right>$ an integral domain?
- Show that the factor ring $\mathbb{Z}[x]/\left<n\right>$ is never a field, regardless of which $n$ you pick.
- We can look at other ideals of $\mathbb{Z}[x]$.
- Show that $\mathbb{Z}[x]/\left<x\right>$ is an integral domain, but not a field.
- Show that $\mathbb{Z}[x]/\left<3,x\right>$ is a field.
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