Can u guys help me prove that $$\mathbb{Q}[x,y]/(x^3+y,x^2+y) \bigotimes_{\mathbb{Q}[x,y]} \mathbb{Q}[x,y]/(x^2-y)$$has dimension 2 as a vector space over $\mathbb{Q}$. I'm not having good ideas about it, thanks.
2026-03-30 16:48:19.1774889299
Proving a tensor product has dimension 2 over $\mathbb{Q}$
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Hint: Use that for a commutative ring $R$ you have $R/I\otimes_R R/J\cong R/(I+J)$, see e.g. this math.stackexchange question.
Then try to find a basis of $R/(I+J)$ by using the conditions in $I+J$. Further hint: In $R/(I+J)$ you have $x^2=y$ and $x^2=-y$. What does that say about $y$?