Let $R=F[X,Y]/(Y^2-X^3)$. Determine the integral closure of $R$ in its quotient field.
I guess I should reduce the problem to some statement related to $F[X]$. For $F$ of characteristic not equal to 2, I proved that the integral closure is $F[X, \sqrt{X}]$. But how to work with the case when $\mathrm{char}(F)=2$?
$R\simeq F[T^2,T^3]$ by $X\to T^2$, $Y\to T^3$. Since the integral closure of $F[T^2,T^3]$ is $F[T]$ we can conclude that the integral closure of $R$ is $R[\frac yx]=F[x,\frac yx]=F[x,\sqrt x]$.