I want to know what is the dimension of this ring $\mathbb C[x,y]_{(0,0)}/(y^2-x^7,y^5-x^3)$.
I don't know how to do that. If I suppose $y^2=x^7$ I will get a higher degree of $x$.
2026-03-29 08:44:36.1774773876
What is the Krull dimension of this local ring
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We compute in the local ring: $$y^{6}=(y^2)^3=(x^7)^3=(x^3)^7=(y^5)^7=y^{35}$$ Hence, $y^6(1-y^{29})=0$. Since $y$ belongs to the maximal ideal, it follows that $1-y^{29}$ is a unit, hence $y^6=0$. Similarly one obtains $x^6=0$. Hence, the local ring modulo nilpotents (this doesn't change dimension) is just $\mathbb{C}$, which is $0$-dimensional.