Is $\mathbb{Q}[X]/(X^3-1)$ isomorphic to $\mathbb Q[X]/(X^3+1)$?

Question

Let $A = \mathbb{Q}[X]/(X^3-1)$.

1. Prove that $A$ is a direct product of two integral domain;
2. Is the ring $A$ isomorphic to $\mathbb{Q}[X]/(X^3+1)$? Justify your answer.

My idea: For question 1, $$ x^3-1 = (x-1)(x^2+x+1),$$ so$$\mathbb{Q}[x]/(x^3-1) = \mathbb{Q}[x]/(x-1) × \mathbb{Q}[x]/(x^2+x+1).$$

For question 2, yes, $A$ isomorphic to $\mathbb{Q}[x]/(x^3+1)$.

Is it correct?