From Dummit & Foote:
I have shown the first part of this exercise: showing $k(t)$ is the fraction field of $R$.
However, I'm not sure how to proceed with the second statement.
We need to show that $k[\frac{\bar x}{\bar y}]$ is a UFD.
2026-04-13 10:43:45.1776077025
Suppose $k$ is a field. Let $\frac{\bar x}{\bar y} \in Frac(k[x,y]/(x^2-y^3))$. How is $k[\frac{\bar x}{\bar y}]$ a UFD?
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Well, $t^2=x^2/y^2=y^3/y^2=y$ in $R$ and $t^3=x^3/y^3=x^3/x^2=x$ in $R$ so $R\subseteq k[t]$. These are two ways of seeing that $t$ is in the integral closure of $R$. Also $k[t]$ is a UFD and so integrally closed.