Krull dimension of this local ring

Question

I want to know what the Krull dimension of this ring $\mathbb C[x,y]_p/(y^2-x^7,y^5-x^3)$ is, where $p\neq (0,0)$. I know the dimension of it in the origin point, but I don't know other cases.

Answer 1

Since $y^2-x^7,y^5-x^3$ are irreducible polynomials (why?) they form a regular sequence in $\mathbb C[x,y]$ and so they are in $\mathbb C[x,y]_{\mathfrak m}$, for any maximal ideal containing them. Then the dimension of $\mathbb C[x,y]_{\mathfrak m}/(y^2-x^7,y^5-x^3)$ is $\dim\mathbb C[x,y]_{\mathfrak m}-2$, that is, $0$.

If $\mathfrak m$ doesn't contain one or both of the polynomials $y^2-x^7,y^5-x^3$, then the quotient ring is $0$.