I can show that $\mathbb{R}(x)[y]/(x^2+y^2-1)$ is a field, but how do we know it is the smallest field containing $\mathbb{R}[x,y]/(x^2+y^2-1)$? I guess we can define the canonical map. Is there any other way to explain the statement?
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2In $\mathbb{R}[x,y]/(x^2+y^2-1)$, $x$ is transcendental over $\Bbb{R}$ thus the fraction field contains $\mathbb{R}(x)$, and $y$ is algebraic with minimal polynomial $x^2+y^2-1$ over it thus $Frac(\mathbb{R}[x,y]/(x^2+y^2-1)) = Frac(\mathbb{R}(x)[y]/(x^2+y^2-1))=\mathbb{R}(x)[y]/(x^2+y^2-1)$ – reuns Aug 20 '19 at 23:59
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3Start with $A[X]/(X^2-a)$, where $A$ is a UFD and $a\in A$ is squarefree, and try to show that its field of fractions is $Q(A)[X]/(X^2-a)$, – user26857 Aug 21 '19 at 10:11