Describe the ring $\mathbb{Z}[x]/I$ for the ideals $I$

Question

Describe to the ring $\mathbb{Z}[x]/I$ for the ideals $I$:

1. $I = (3)$
2. $I = (x)$
3. $I = (x^2)$
4. $I = (x^2+1)$
5. $I = (3(x^2+1))$

I managed to show the following points by using the fundamental theorem on homomorphisms:

• $\mathbb{Z}[x]/(3) \cong \mathbb{Z}_3[x]$
• $\mathbb{Z}[x]/(x) \cong \mathbb{Z}$
• $\mathbb{Z}[x]/(x^2+1) \cong \mathbb{Z}[\sqrt{-1}]$

For the third part I have the problem that that $x^2$ has the same root as $x$ so I don't really know how to approch this case. My intuition would tell me that $\mathbb{Z}[x]/(x^2)$ is isomorphic to all linaer polynomial in $\mathbb{Z}[x]$.

For $\mathbb{Z}[x]/(3(x^2+1))$ I tried to use one of the isomorphism theorems to connect the results from (1) and (4).

I am trying to solve this exercise to prepare for an algebra exam. I would really appreciate some help - maybe there is a general way to approach these problems.

Answer 1

Let $\;L:=\{p(x)\in\Bbb Z[x]\;/\;\deg p\le 1\}\;$ with the operations inherited from $\;\Bbb Z[x]\;$, and define

$$\phi:\Bbb Z[x]\to L\;,\;\;\phi\left(\sum_{k=0}^n a_k x^k\right):=a_1x+a_0$$

Show the above is an epimorphism of rings, and $\;\ker\phi=\langle x^2\rangle\;$