Let $\mathbb{Q}$ be the rational numbers, and let $k = \mathbb{Q}[X]/(X^3-2)$. Describe the $\mathbb{Q}$-algebra $k \otimes_{\mathbb{Q}} k$ as explicitly as possible. Is it a domain?
We have been told to use the fact that, for any R-algebra $S$ ($\phi: R \to S), $
$$R[X]/f(x) \otimes_R S = S[X]/f^{\phi}(x)$$ where $f^{\phi}(x)$ denotes the polynomial obtained by applying $\phi$ to the coefficients of $f(x)$.
Using this, I can see that $k \otimes_{\mathbb{Q}} k \equiv k[Y]/(Y^3-2)$. But I am not sure how to proceed in the problem.