I want to find all ideals in $\mathbb{Q}[x]/I$, where $I$ is the ideal generated by $p(x)=x^2(x^2+x+1)$. I know that $$\mathbb{Q}[x]/I \cong \mathbb{Q}[x]/(x^2) \times \mathbb{Q}[x]/(x^2+x+1)$$
I have found three ideals so far:
$1) J_1=$Ideal generated by $(1,0)$ which is prime and maximal
$2)J_2=$ Ideal generated by $(x,0)$ which is prime but not maximal, since $J_2 \subsetneq J_1 \subsetneq \mathbb{Q}[x]/I $
$3)J_3=$ Ideal generated by $(0,1)$ which is prime and maximal
Note that Ideal generated by $(0,x)$ (denote it by $J_4$) is exactly $J_3$, because : $$ J_4 \ni (0,x)\cdot(0,x)+(0,x)=(0,x^2)+(0,x)=(0,-1-x)+(0,x)=(0,-1)$$ Hence $J_3=J_4$.
My question is : Are there any other ideals in $\mathbb{Q}[x]/I$ ? If yes are they prime/maximal ?
I would be thankful if you verify the following proof of the question : "is $\mathbb{Q}[x]/(x^2+x+1)$ a field or not?"
Since $\mathbb{Q}[x]$ is a Euclidean domain , and hence a P.I.D, and since $x^2+x+1$ is irreducible in $\mathbb{Q}[x]$(because it has no roots in $\mathbb{Q}$) $(x^2+x+1)$ is prime and maximal, Therefore $\mathbb{Q}[x]/(x^2+x+1)$ is a field.
The ideals in $\mathbb{Q}[x]/I$ correspond to the ideals in $\mathbb{Q}[x]$ that contain $I$.
These are principal ideals generated by the divisors of $x^2(x^2+x+1)$.
There are $(2+1)\cdot(1+1)$ such divisors. Find them.
This is similar to finding the ideals of $\mathbb{Z}/I$ when $I=(p^2 q)$, where $p$ and $q$ are prime numbers.