Let $\displaystyle K=\mathbb{Q}[\omega]$ where $\displaystyle \omega^2+\omega+1=0$ and let $\displaystyle R$ be the polynomial ring $\displaystyle K[x]$. Let $\displaystyle L$ be the field $\displaystyle K(x)[y]$ where $\displaystyle y$ satisfies $\displaystyle y^3=1+x^2$.

Which is the integral closure of $\displaystyle R$ in $\displaystyle L$, why?