Can you show me how to compute : $ \mathrm{dim}_{ \mathbb{C} } \big( \mathbb{C} [X,Y] / ( Y^2 - X^2 , Y^2 + X^2 ) \big) $ ?
Thank you in advance.
2026-05-03 15:20:10.1777821610
What is the dimension over $ \mathbb{C} $ of $ \mathbb{C} [X,Y] / ( Y^2 - X^2 , Y^2 + X^2 ) $?
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Clearly in the quotient ring $Y^2=0=X^2$. The quotient ring is generated as a complex vector space by all monomials in $X$ and $Y$; with the above fact it is clear now that $1,X,Y,XY$ will form a basis, and so the dimension is $4$.