I'm trying to show that $\mathbb{Q} [x ] / ( x^3-2)$ is not isomorphic to $\mathbb{Q} [x ] / ( x^3-3)$.

My attempt : Since $\mathbb{Q} [x ] / ( x^3-2)$ is isomorphic to $\mathbb{Q} [2^{1/3}]$ and $\mathbb{Q} [x ] / ( x^3-3)$ is isomorphic to $\mathbb{Q} [3^{1/3}]$, I will show that $\mathbb{Q} [2^{1/3}]$ is not isomorphic to $\mathbb{Q} [3^{1/3}].$ Is it correct idea?