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I am reading Reid "Undergraduate Commutative Algebra". At some point (page 62) he's talking about the ring $k[X,Y]/(Y^2-X^3)$, where $k$ is a field. He claims that it is easy to see that its ring of fractions is $k[t]$, where $t=y/x$ and lowercase letters indicate equivalence classes modulo $(Y^2-X^3)$ (so $x$ is the equivalence class of $X$ modulo the ideal). Can someone explain how to see $k[t]$ is the ring of fractions?

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  • $\begingroup$ Are you sure it's supposed to be $k[t]$? It seems to me that it is not a field. $k(t)$ would be fine. $\endgroup$ – tomasz Nov 7 '17 at 0:52
  • $\begingroup$ Yes, it says verbatim "it is not hard to see that it has field of fractions $\mathrm{Frac}A=k(t)$ ". I must say I just noticed that here he uses round brackets and not square brackets, but I do not quite understand the difference in the two notations $k[t]$ and $k(t)$ either (a $\endgroup$ – Enrico Nov 7 '17 at 1:00
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    $\begingroup$ $k[t]$ is the smallest ring containing $k$ and $t$. $k(t)$ is the smallest field. $\endgroup$ – tomasz Nov 7 '17 at 1:02
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Hint: Notice that $t^2=x$. Conclude that $x,y\in k(t)$.

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