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The question arises from reading the answer here. $k$ here is a field.

The proof reads, "But $k[X,Y,T]/(Y^2-X^2(X-1),XT-Y,T^2-X+1)\simeq k[T]$, and thus we get two things..."

My Question: I can't quite see why the two sides above are isomorphic. Any help would be greatly appreciated.

user26857
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Dick Grayson
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1 Answers1

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We have the following chain of isomorphisms: \begin{align*} k[X,Y,T]/(Y^2-X^2(X-1),XT-Y,T^2-X+1) &\cong k[T^2+1,Y,T]/(Y^2-(T^2+1)^2T^2,(T^2+1)T-Y),\text{ imposed by the element $T^2-X+1$}\\ &=k[Y,T]/(Y^2-(T^2+1)T^2,(T^2+1)T-Y)\\ &\cong k[(T^2+1)T,T]/((T^2+1)^2T^2-(T^2+1)^2T^2),\text{ imposed by the element $(T^2+1)T-Y$}\\ &=k[T]/(0)\\ &\cong k[T] \end{align*}

Fotis
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