fraction field of the integral closure
Let $R$ be a domain, $K$ the field of fractions of $R$ and $L$ a finite field extension of $K$. Denote with $R'$ the integral closure of $R$ in $L$. Is it always true that $L$ is the field of fractions of $R'$? If not, is there an easy counter-example?
It is true in general. If $x \in L$, then it is the root of some non-zero polynomial over $R$ (since $x$ is algebraic over $K$, we just clear denominators). Let $a$ be the leading coefficient. Then one checks that $ax$ is integral over $R$. Hence, $ax \in R'$. This shows $L=Q(R')$.