Let $R$ be a polynomial ring of several variables over $\mathbb{Q}$ (says $x_1,...,x_n$) and $I$ be a prime ideal of $k$. We already knew that $R$ is an integrally closed domain since it is a UFD. Is being integrally closed also applied for the quotient ring $R/I$?
If the statement is not true in general, let us consider the specific case $n=3$ and $I=(x_1^3-x_2^6x_3)$. I have already checked that $I$ is a prime ideal using Eisenstein's criterion. Now is $R/I$ integrally closed? I knew some tools that can be used to check the integral property of a ring extension, but so far I don't know any tools than can also be apply for these cases, since we are working with the integrally closed property over the fraction field.
Any help is appreciated.
The first question asks if every integral domain which is finitely generated over $\mathbb{Q}$ is integrally closed. This is not correct (of course, otherwise we would not need that notion). For example, $\mathbb{Q}[T^2,T^3] \cong \mathbb{Q}[X,Y]/\langle Y^2-X^3\rangle$ is not integrally closed (consider $a=Y/X$, then $a^2=X$).
For your other example $A = \mathbb{Q}[X,Y,Z]/\langle X^3-Y^6 Z \rangle$, in $Q(A)$ we have $ZY^3=(X/Y)^3$, and you can prove $X/Y \notin A$. Thus, $A$ is not integrally closed.