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The question is this: Let $f:C[x,y]\rightarrow C[t]$ 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 $C[x,y]$ 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 $C[x,y]/K \to C[t]$. –  Carl Aug 29 '12 at 6:40
    
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?? –  Briana Aug 29 '12 at 7:20
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1 Answer

We know that $C[x,y]/K \cong C[t]$, and that $C[t]$ is a PID, being the ring of polynomials in one variable over a field. Thus, $C[x,y]/K$ is also a PID. Therefore, if $I$ is any ideal containing $K$, we know that $I/K$ is principle in $C[x,y]/K$, so it can be written $(\bar{z})$ where $\bar{z}$ is the coset of some element $z \in C[x,y]$. It remains to check that $z$ and $(x-1)^3-y-1$ generate $I$. This would show that $I$ is generated by at most two elements, which is what the problem asks.

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Thank you for your answer! –  Briana Aug 29 '12 at 18:29
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