Prime ideals of the ring $\mathbb{Q}[X]/ (X)^3$

Question

The following question was part of my abstract algebra quiz and I am unable to solve it.

Let A denote the ring $\mathbb{Q}[X]/ (X)^3$ . Then is there only 1 prime ideal of A?

Ideals of $A$ are $x/\langle x^3 \rangle$ , $x^2 / \langle x^3 \rangle$ and $1$ and $ 1/\langle x^3 \rangle$ .

I proved $x/\langle x^3 \rangle$ prime. THen I tried with $x^2 / \langle x^3 \rangle$. Here any element of $A/(x^2 / \langle x^3 \rangle)$ would be of the form $a_0 + a_1 x$ but I am neither able to prove it prime or not prime.

As other ideals would be of the same form so I am looking for you help to do this part.

Answer 1

$A/((X)/(X^3))=(\mathbb{Q}[X]/(X^3))/((X^2)/(X^3))\cong \mathbb{Q}[X]/(X^2)$.

Now $X\neq 0\in \mathbb{Q}[X]/(X^2)$ and $X\cdot X=0\in \mathbb{Q}[X]/(X^2)$, so $\mathbb{Q}[X]/(X^2)$ is not integral domain.