Determining the nature of these rings How can we determine whether the quotient rings $R/I$ where where $R=\mathbb Z[x], \mathbb Z_2[x], \mathbb Z_3[x]$ and $I=(x^2+1), (x^2+x+1)$ are fields, principal ideal domains, unique factorization domains or integral domain? (Of course they are all integral domains, but I am looking for the strictest condition.) Thank you.
 A: Note that $R/I$ is a field if and only if $I$ is maximal and $R/I$ is an integral domain if and only if $I$ is prime.
Also note that if $F$ is a field then $(p(x))$ is maximal in $F[x]$ if and only if $p(x)$ is irreducible over $F$.
Note that $\mathbb{Z}_2$ and $\mathbb{Z}_3$ are fields. Now use that $f(x) \in F[x]$ with $\operatorname{deg}{(f)} \in \{2,3\}$ is reducible if and only if it has a root in $F$. 
First let $p(x) = x^2 + 1$ and $R = \mathbb{Z}_3 [x]$. Note that $p(0) = 1 \neq 0$, $p(1) = 2 \neq 0$ and $p(2) = 2 \neq 0$. Hence $p(x)$ is irreducible and therefore $I = (p(x))$ is maximal and therefore $R/I$ is a field in this case. Every field is also an integral domain, a PID and a UFD, so you're done.
Now let $p(x) = x^2 + x + 1$ and $R = \mathbb{Z}_2[x]$. Is $p(x)$ irreducible over $R$? 
Hope this helps. I'll add some more details later.
A: Her are some (disorganized) ramblings.
First of all: they are not all integral domains. For example, $x^2+1$ is reducible modulo 2, namely as $(x+1)^2$, so $\mathbb{Z}_2[x]/(x^2+1)$ is not an integral domain.
Similarly, $x^2+x+1=x^2+x+1-3x=(x-1)^2$ mod 3, so $\mathbb{Z}_3[x]/(x^2+x+1)$ is not an integral domain either.
For the other case, note that since $\mathbb{Z}_2$ and $\mathbb{Z}_2$ are fields, $\mathbb{Z}_2[x]$ and $\mathbb{Z}_3[x]$ are PID's. Hence so are their quotient rings.
Note also that $\mathbb{Z}[x]/(x^2+1) \cong \mathbb{Z}[i]$. 
