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Is ring $\mathbb{C}[T_1, T_2]/(T_2^2 - T_1^3 - 1)$ isomorphic to $\mathbb{C}[T, \sqrt{T^3+1}]$? I know the first one is the ring of polynomial functions defined on curve $y^2 = x^3 + 1$.

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Can t you see how to define maps from one to the other which are inverse isomorphisms? – Mariano Suárez-Alvarez Jan 11 '13 at 6:33
I see that map, which sends $T_1$ to $T$, $T_2$ to $\sqrt{T^3+1}$. But I don't know how to prove it formally that it is an isomorphism. – user42212 Jan 11 '13 at 6:53
Find a map in the other direction and check that the two compositions are identities. – Mariano Suárez-Alvarez Jan 11 '13 at 7:05
What do you think $\mathbb{C}[T, \sqrt{T^3 + 1}]$ means if it doesn't mean "$\mathbb{C}[T]$ adjoin a thing which squares to $T^3 + 1$" (namely $T_2$ in the first ring)? – Qiaochu Yuan Jan 11 '13 at 7:27
@QiaochuYuan A possible answer to your question is that user42212 has chosen a nonempty open subset $\Omega$ of $\mathbb{C}$, and a branch of $\sqrt{T^3+1}$ on $\Omega$, and then wants to study the ring of functions on $\Omega$ generated by $T$ and $\sqrt{T^3+1}$. (This is analogous to how some people would define $\mathbb{Q}[\sqrt{2}]$ as a subfield of $\mathbb{R}$, rather than by formally adjoining a root of $2$ to $\mathbb{Q}$.) But, in any case, I agree that user42212 should think through and explain what he or she means by $\mathbb{C}[T, \sqrt{T^3+1}]$. – David Speyer Jan 11 '13 at 17:07

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