Field of fractions of $\mathbb{Q}/\langle x^2+y^2-1\rangle$

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  • $\mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain, and its field of fractions is isomorphic to $\mathbb Q(t)$

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This follows from the classification of Pythagorean triples.

A little bit more detailed: The rational solutions of $x^2+y^2=1$ are parametrized $x=\frac{2t}{1+t^2}$ and $y=\frac{1-t^2}{1+t^2}$. But actually the equation $\left(\frac{2t}{1+t^2}\right)^2 + \left(\frac{1-t^2}{1+t^2}\right)^2=1$ holds formally in $\mathbb{Q}(t)$. Thus, there is a homomorphism $R=\mathbb{Q}[x,y]/(x^2+y^2-1) \to \mathbb{Q}(t)$ given by $x \mapsto \frac{2t}{1+t^2}$ and $y \mapsto \frac{1-t^2}{1+t^2}$. Verify that it is injective, hence extends to a homomorphism $\mathrm{Quot}(R) \to \mathbb{Q}(t)$. Check that $t$ lies in the image. Hence, this homomorphism is an isomorphism.