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Let $k$ be a field, consider the ring $ k[X,Y]/(X^2-Y^3) $ I was proving something but I need to prove the existence of an element in the ring of fractions of $ k[X,Y]/(X^2-Y^3) $ such that satisfy a monic polynomial with coefficients in $ k[X,Y]/(X^2-Y^3) $ , but an element that does not belong to the ring $ k[X,Y]/(X^2-Y^3) $. How Can I find that element?

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You need to take advantage of the fact that the curve $X^2=Y^3$ has a singularity at the origin. Hint: try something simple divided by $X$ or $Y$. Another hint ("orthogonal" to the first hint): Have you seen a rational parametrization of this curve? –  Jyrki Lahtonen Mar 27 '12 at 6:42
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The element $x/y\in \operatorname {Frac} (k[x,y])=\operatorname {Frac}(k[X,Y]/(X^2-Y^3)) \:$ satisfies $T^2-y=0$.

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