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I have to prove one of the following (equivalent) statements: that the ideal $J=\langle Y^{3}-X,Z^{5}-Y^{4}\rangle$ is prime or that $\mathbb{C}[X,Y,Z]/J$ is an integral domain.

My teacher told me that proving that $\mathbb{C}[X,Y,Z]/J$ is an integral domain is easier, but I don't know where to start.

I thought maybe I could find an isomorphism to a well known integral domain, but the fact that it is generated by two polynomials and that there is no easy parametrization is making this difficult to me.

I tried an isomorphism to $\mathbb C [X, X^{1/3}, X^{4/15}]$, but he asked me what meaning $X^{1/3}$ and $X^{4/15}$ had (we never used polynomial rings with fractional powers in this subject) and I didn't know what to answer.

So maybe there is another (easier) way that I didn't thought.

Miguel Mars
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2 Answers2

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Don't use $\Bbb C[X,X^{1/3},X^{4/15}]$, use $\Bbb C[T^{15},T^5,T^4]$ (a subring of $\Bbb C[T]$). She can't complain about that. Map $X$ to $T^{15}$, $Y$ to $T^5$ and $Z$ to $T^4$. Then all you have to do is to show the kernel is generated by $X-Y^3$ and $Y^4-Z^5$.

Angina Seng
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In the quotient we have $X=Y^3$, so $X$ is redundant, i.e. your ring is isomorphic to $\mathbb{C}[Y,Z]/(Z^5-Y^4).$ Since $\mathbb{C}[Y,Z]$ is a UFD, you need to show $Z^5-Y^4$ is irreducible. This is true because $4$ and $5$ are coprime (see here: Necessary and sufficient condition for $x^n - y^m$ to be irreducible in $\Bbb C[x,y]$).

anon
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