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The question is this:

Let $f:\mathbb C[x,y]\rightarrow\mathbb C[t]$ be the homomorphism that sends $x\mapsto t+1$ and $y\mapsto t^3-1.$ Determine the kernel $K$ of $f$, and prove that every ideal $I$ of $\mathbb C[x,y]$ that contains $K$ can be generated by two elements.

Solution:

I have shown that $((x-1)^3-y-1)=K.$ However, I am having trouble showing the second part. Any hints??

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    I assume $C$ is supposed to be the complex numbers, or at least a field? In that case, consider that $C[t]$ is a PID, and that as $f$ is surjective, it induces an isomorphism $C[x,y]/K \to C[t]$.2012-08-29
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    Yes $C$ is the complex field. So does this mean then, that if $I$ is an ideal in $C[x,y]$, then $I$ will also be an ideal in $C[x,y]/K$ and since this is isomorphic to $C[t]$ and all ideals are principal here, and can be generated by more than one element, then I is generated by more than one??2012-08-29

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