I'm thinking about this question and no elegant idea comes to my mind. Prove that the field of fractions of $\mathbb R[x,y]/(x^2+y^2-1)$ is isomorphic to $\mathbb R (t)$.
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1Are you sure you want $\mathbb R(t)$ and not $\mathbb R[t]$? – Alex Becker Aug 20 '13 at 03:49
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@AlexBecker yes. I changed the word "quotients" by "fractions" to avoid ambiguities – user91050 Aug 20 '13 at 03:56
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Ah, that's what I'd misunderstood. – Alex Becker Aug 20 '13 at 03:56
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3use a rational parameterization of the circle – yoyo Aug 20 '13 at 04:18
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2Dear user, This answer is closely related. Regards, – Matt E Aug 20 '13 at 04:28
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@MattE - Thanks! It works. I was thinking about the properties of $\mathbb (R[x])[y]/(y^2+(x^2-1))$, but in fact got lost. – user91050 Aug 20 '13 at 04:52
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2I strongly believe this is a duplicate, but I can't track down the previous question at the moment. – Potato Aug 20 '13 at 06:22
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2If the field of fractions is isomorphic to $\mathbb{R}(t)$, then the ring itself must embed into $\mathbb{R}(t)$. Finding an embedding into $\mathbb{R}(t)$ involves finding a pair of rational functions $x(t), y(t)$ such that $x(t)^2 + y(t)^2 = 1$. But this is precisely a rational parameterization of the circle. – Qiaochu Yuan Aug 20 '13 at 07:03
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1Duplicate of http://math.stackexchange.com/questions/373504/field-of-fractions-of-mathbbqx-y-langle-x2y2-1-rangle which is in turn a duplicate of http://math.stackexchange.com/questions/96400 – Martin Brandenburg Aug 20 '13 at 07:39