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I am trying to prove that the ideal in ${\mathbb C}[x,y,z]$ generated by $z-x^3$ and $y-x^2$ is prime. I know I could take a suitable quotient and show it's domain. But I am rather stuck.

Math101
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1 Answers1

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Hint: Consider ${\mathbb C}[x,y,z] \to {\mathbb C}[x]$ induced by $x\mapsto x$, $y\mapsto x^2$, $z\mapsto x^3$.

lhf
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  • I see...the map is surjective and the kernel is $(z-x^3,y-x^2)$, so the quotient is isomorphic to ${\mathbb C}[x]$, that is a domain. – Math101 Oct 12 '19 at 08:22