prove that every root closed domain is integrally closed
I want to help but I am not sure what you mean by "root closed domain" and by "integrally closed domain". Here is my understanding, if $\displaystyle R$ is a subsring of a domain $\displaystyle D$ and $\displaystyle R$ is root closed, i.e. every element in $\displaystyle R$ is a zero of some (monic?) polynomial. Then it is integrally closed meaning if $\displaystyle d\in D$ then it is zero of some (monic?) polynomial in $\displaystyle R$. Right or wrong?