I hope that the following questions can be solved by Ring Theory concepts as I'm not studying further yet.
Let $A$ be an integral domain and $K$ its field of fractions. An integral domain $A$ is said to be integrally closed if $A$ satisfies the following condition: for $\alpha \in K$ if there exists a monic polynomial $f(X)$ whose coefficients are in $A$ satisfying $f(\alpha)=0$, then $\alpha \in A$.
My first question is to prove that if an integral domain $A$ is integrally closed, then the ring of fractions $S^{-1}A$ of $A$ with respect to a multiplicatively closed subset $S$ of $A$ is also integrally closed.
My attempt: Since $A$ is integrally closed, we have the integral closure of $A$ is $A$ itself. It is a theorem that (see here for instance) $S^{-1}A$ is the integral closure of $S^{-1}A$ in $S^{-1}K=K$. It follows that $S^{-1}A$ is also integrally closed.
Is this a correct proof?
My second question is trying to check whether each of the following integral domains $A$ is integrally closed or not.
(a) $A=\mathbb C[X,Y]/(X^{n}-Y^{m})$, where $n>m$ be coprime positive integers.
(b) $A=\mathbb C[X,Y]/(XY-1)$.
I'm might be aware of events that both of these above integral domains are not UFDs but I have no idea about their integrally closedness. Any help would be much appreciated
(a) $\mathbb C[X,Y]/(X^{n}-Y^{m})\simeq\mathbb C[T^m,T^n]$, where $n>m$ are coprime positive integers. The field of fractions of $\mathbb C[T^m,T^n]$ is $\mathbb C(T)$ (why?), $T$ is integral over $\mathbb C[T^m,T^n]$, and $T\notin\mathbb C[T^m,T^n]$.
(b) $\mathbb C[X,Y]/(XY-1)\simeq S^{-1}\mathbb C[X]$, where $S=\{1,X,X^2,\dots\}$.